elrspunsn - Mathbox for Thierry Arnoux

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Theorem elrspunsn 32398

Description: Membership to the span of an ideal 𝑅 and a single element 𝑋. (Contributed by Thierry Arnoux, 9-Mar-2025.)

**Hypotheses**
Ref	Expression
elrspunidl.n	⊢ 𝑁 = (RSpan‘𝑅)
elrspunidl.b	⊢ 𝐵 = (Base‘𝑅)
elrspunidl.1	⊢ 0 = (0_g‘𝑅)
elrspunidl.x	⊢ · = (._r‘𝑅)
elrspunidl.r	⊢ (𝜑 → 𝑅 ∈ Ring)
elrspunsn.p	⊢ + = (+_g‘𝑅)
elrspunsn.i	⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
elrspunsn.x	⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝐼))

**Assertion**
Ref	Expression
elrspunsn	⊢ (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))

Distinct variable groups: 0 ,𝑖 · ,𝑖 𝐵,𝑖 𝑅,𝑖 𝑖,𝑋 𝜑,𝑖 + ,𝑖,𝑟 0 ,𝑟 · ,𝑟 𝐴,𝑖,𝑟 𝐵,𝑟 𝑖,𝐼,𝑟 𝑅,𝑟 𝑋,𝑟 𝜑,𝑟 Allowed substitution hints: 𝑁(𝑖,𝑟)

Proof of Theorem elrspunsn Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrspunidl.n	. . 3 ⊢ 𝑁 = (RSpan‘𝑅)
2		elrspunidl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		elrspunidl.1	. . 3 ⊢ 0 = (0_g‘𝑅)
4		elrspunidl.x	. . 3 ⊢ · = (._r‘𝑅)
5		elrspunidl.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		elrspunsn.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))
7		eqid 2731	. . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
8	2, 7	lidlss 20781	. . . . 5 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵)
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ 𝐵)
10		elrspunsn.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝐼))
11	10	eldifad 3956	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12	11	snssd 4805	. . . 4 ⊢ (𝜑 → {𝑋} ⊆ 𝐵)
13	9, 12	unssd 4182	. . 3 ⊢ (𝜑 → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
14	1, 2, 3, 4, 5, 13	elrsp 32348	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))))))
15		oveq1 7400	. . . . . . . 8 ⊢ (𝑟 = (𝑎‘𝑋) → (𝑟 · 𝑋) = ((𝑎‘𝑋) · 𝑋))
16	15	oveq1d 7408	. . . . . . 7 ⊢ (𝑟 = (𝑎‘𝑋) → ((𝑟 · 𝑋) + 𝑖) = (((𝑎‘𝑋) · 𝑋) + 𝑖))
17	16	eqeq2d 2742	. . . . . 6 ⊢ (𝑟 = (𝑎‘𝑋) → (𝐴 = ((𝑟 · 𝑋) + 𝑖) ↔ 𝐴 = (((𝑎‘𝑋) · 𝑋) + 𝑖)))
18		oveq2 7401	. . . . . . 7 ⊢ (𝑖 = (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) → (((𝑎‘𝑋) · 𝑋) + 𝑖) = (((𝑎‘𝑋) · 𝑋) + (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)))))
19	18	eqeq2d 2742	. . . . . 6 ⊢ (𝑖 = (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) → (𝐴 = (((𝑎‘𝑋) · 𝑋) + 𝑖) ↔ 𝐴 = (((𝑎‘𝑋) · 𝑋) + (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))))))
20		elmapi 8826	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋})) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
21	20	ad3antlr 729	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
22	11	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝑋 ∈ 𝐵)
23		snidg 4656	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ {𝑋})
24		elun2 4173	. . . . . . . 8 ⊢ (𝑋 ∈ {𝑋} → 𝑋 ∈ (𝐼 ∪ {𝑋}))
25	22, 23, 24	3syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝑋 ∈ (𝐼 ∪ {𝑋}))
26	21, 25	ffvelcdmd 7072	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑎‘𝑋) ∈ 𝐵)
27	2	fvexi 6892	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
28	27	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝐵 ∈ V)
29	6	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝐼 ∈ (LIdeal‘𝑅))
30		ssun1 4168	. . . . . . . . . . . 12 ⊢ 𝐼 ⊆ (𝐼 ∪ {𝑋})
31	30	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝐼 ⊆ (𝐼 ∪ {𝑋}))
32	21, 31	fssresd 6745	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑎 ↾ 𝐼):𝐼⟶𝐵)
33	28, 29, 32	elmapdd 8818	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑎 ↾ 𝐼) ∈ (𝐵 ↑_m 𝐼))
34		breq1 5144	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑎 ↾ 𝐼) → (𝑏 finSupp 0 ↔ (𝑎 ↾ 𝐼) finSupp 0 ))
35		fveq1 6877	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ↾ 𝐼) → (𝑏‘𝑦) = ((𝑎 ↾ 𝐼)‘𝑦))
36	35	oveq1d 7408	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ↾ 𝐼) → ((𝑏‘𝑦) · 𝑦) = (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦))
37	36	mpteq2dv 5243	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑎 ↾ 𝐼) → (𝑦 ∈ 𝐼 ↦ ((𝑏‘𝑦) · 𝑦)) = (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦)))
38	37	oveq2d 7409	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑎 ↾ 𝐼) → (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ ((𝑏‘𝑦) · 𝑦))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦))))
39	38	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑎 ↾ 𝐼) → ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ ((𝑏‘𝑦) · 𝑦))) ↔ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦)))))
40	34, 39	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑏 = (𝑎 ↾ 𝐼) → ((𝑏 finSupp 0 ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ ((𝑏‘𝑦) · 𝑦)))) ↔ ((𝑎 ↾ 𝐼) finSupp 0 ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦))))))
41	40	adantl 482	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) ∧ 𝑏 = (𝑎 ↾ 𝐼)) → ((𝑏 finSupp 0 ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ ((𝑏‘𝑦) · 𝑦)))) ↔ ((𝑎 ↾ 𝐼) finSupp 0 ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦))))))
42		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝑎 finSupp 0 )
43	5	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝑅 ∈ Ring)
44	2, 3	ring0cl 20041	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵)
45	43, 44	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 0 ∈ 𝐵)
46	42, 45	fsuppres 9371	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑎 ↾ 𝐼) finSupp 0 )
47		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (𝑎‘𝑥) = (𝑎‘𝑦))
48		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
49	47, 48	oveq12d 7411	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑎‘𝑥) · 𝑥) = ((𝑎‘𝑦) · 𝑦))
50	49	cbvmptv 5254	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)) = (𝑦 ∈ 𝐼 ↦ ((𝑎‘𝑦) · 𝑦))
51		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
52	51	fvresd 6898	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) ∧ 𝑦 ∈ 𝐼) → ((𝑎 ↾ 𝐼)‘𝑦) = (𝑎‘𝑦))
53	52	oveq1d 7408	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) ∧ 𝑦 ∈ 𝐼) → (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦) = ((𝑎‘𝑦) · 𝑦))
54	53	mpteq2dva 5241	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦)) = (𝑦 ∈ 𝐼 ↦ ((𝑎‘𝑦) · 𝑦)))
55	50, 54	eqtr4id 2790	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)) = (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦)))
56	55	oveq2d 7409	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦))))
57	46, 56	jca 512	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → ((𝑎 ↾ 𝐼) finSupp 0 ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ (((𝑎 ↾ 𝐼)‘𝑦) · 𝑦)))))
58	33, 41, 57	rspcedvd 3611	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → ∃𝑏 ∈ (𝐵 ↑_m 𝐼)(𝑏 finSupp 0 ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ ((𝑏‘𝑦) · 𝑦)))))
59	9	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝐼 ⊆ 𝐵)
60	1, 2, 3, 4, 43, 59	elrsp 32348	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) ∈ (𝑁‘𝐼) ↔ ∃𝑏 ∈ (𝐵 ↑_m 𝐼)(𝑏 finSupp 0 ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑦 ∈ 𝐼 ↦ ((𝑏‘𝑦) · 𝑦))))))
61	58, 60	mpbird 256	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) ∈ (𝑁‘𝐼))
62	1, 7	rspidlid 32349	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑁‘𝐼) = 𝐼)
63	5, 6, 62	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝐼) = 𝐼)
64	63	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑁‘𝐼) = 𝐼)
65	61, 64	eleqtrd 2834	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) ∈ 𝐼)
66		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))))
67		elrspunsn.p	. . . . . . . . . 10 ⊢ + = (+_g‘𝑅)
68	5	ringcmnd 20058	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ CMnd)
69	68	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑅 ∈ CMnd)
70	6	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝐼 ∈ (LIdeal‘𝑅))
71		snex 5424	. . . . . . . . . . . 12 ⊢ {𝑋} ∈ V
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → {𝑋} ∈ V)
73	70, 72	unexd 7724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∪ {𝑋}) ∈ V)
74	5	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑅 ∈ Ring)
75	20	ad3antlr 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
76		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
77	75, 76	ffvelcdmd 7072	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (𝑎‘𝑥) ∈ 𝐵)
78	13	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
79	78, 76	sseldd 3979	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥 ∈ 𝐵)
80	2, 4, 74, 77, 79	ringcld 20037	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → ((𝑎‘𝑥) · 𝑥) ∈ 𝐵)
81	73	mptexd 7210	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)) ∈ V)
82	5, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ∈ 𝐵)
83	82	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 0 ∈ 𝐵)
84		funmpt 6575	. . . . . . . . . . . 12 ⊢ Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))
85	84	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))
86		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑎 finSupp 0 )
87	86	fsuppimpd 9352	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑎 supp 0 ) ∈ Fin)
88	20	ad3antlr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
89	88	ffnd 6705	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑎 Fn (𝐼 ∪ {𝑋}))
90	73	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → (𝐼 ∪ {𝑋}) ∈ V)
91	5	ad3antrrr 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑅 ∈ Ring)
92	91, 44	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 0 ∈ 𝐵)
93		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 )))
94	89, 90, 92, 93	fvdifsupp 31777	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → (𝑎‘𝑥) = 0 )
95	94	oveq1d 7408	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → ((𝑎‘𝑥) · 𝑥) = ( 0 · 𝑥))
96	13	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
97	93	eldifad 3956	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
98	96, 97	sseldd 3979	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → 𝑥 ∈ 𝐵)
99	2, 4, 3	ringlz 20064	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ( 0 · 𝑥) = 0 )
100	91, 98, 99	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → ( 0 · 𝑥) = 0 )
101	95, 100	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 ))) → ((𝑎‘𝑥) · 𝑥) = 0 )
102	101, 73	suppss2 8167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)) supp 0 ) ⊆ (𝑎 supp 0 ))
103	87, 102	ssfid 9250	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)) supp 0 ) ∈ Fin)
104	81, 83, 85, 103	isfsuppd 9349	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)) finSupp 0 )
105	10	eldifbd 3957	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐼)
106	105	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ¬ 𝑋 ∈ 𝐼)
107		disjsn 4708	. . . . . . . . . . 11 ⊢ ((𝐼 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐼)
108	106, 107	sylibr 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∩ {𝑋}) = ∅)
109		eqidd 2732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∪ {𝑋}) = (𝐼 ∪ {𝑋}))
110	2, 3, 67, 69, 73, 80, 104, 108, 109	gsumsplit2 19756	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))) = ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) + (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ ((𝑎‘𝑥) · 𝑥)))))
111	69	cmnmndd 19636	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑅 ∈ Mnd)
112	11	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋 ∈ 𝐵)
113	5	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑅 ∈ Ring)
114	20	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
115		ssun2 4169	. . . . . . . . . . . . . 14 ⊢ {𝑋} ⊆ (𝐼 ∪ {𝑋})
116	11, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
117	116	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋 ∈ {𝑋})
118	115, 117	sselid 3976	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝑋 ∈ (𝐼 ∪ {𝑋}))
119	114, 118	ffvelcdmd 7072	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑎‘𝑋) ∈ 𝐵)
120	2, 4, 113, 119, 112	ringcld 20037	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑎‘𝑋) · 𝑋) ∈ 𝐵)
121		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
122	121	fveq2d 6882	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → (𝑎‘𝑥) = (𝑎‘𝑋))
123	122, 121	oveq12d 7411	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 = 𝑋) → ((𝑎‘𝑥) · 𝑥) = ((𝑎‘𝑋) · 𝑋))
124	2, 111, 112, 120, 123	gsumsnd 19779	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ ((𝑎‘𝑥) · 𝑥))) = ((𝑎‘𝑋) · 𝑋))
125	124	oveq2d 7409	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) + (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ ((𝑎‘𝑥) · 𝑥)))) = ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) + ((𝑎‘𝑋) · 𝑋)))
126	5	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Ring)
127	20	ad3antlr 729	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑎:(𝐼 ∪ {𝑋})⟶𝐵)
128		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
129	30, 128	sselid 3976	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
130	127, 129	ffvelcdmd 7072	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑎‘𝑥) ∈ 𝐵)
131	9	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → 𝐼 ⊆ 𝐵)
132	131, 128	sseldd 3979	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐵)
133	2, 4, 126, 130, 132	ringcld 20037	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ 𝐼) → ((𝑎‘𝑥) · 𝑥) ∈ 𝐵)
134	133	fmpttd 7099	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)):𝐼⟶𝐵)
135	30	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → 𝐼 ⊆ (𝐼 ∪ {𝑋}))
136	135	ssdifd 4136	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝐼 ∖ (𝑎 supp 0 )) ⊆ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 )))
137	136	sselda 3978	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ (𝑎 supp 0 )))
138	137, 94	syldan 591	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → (𝑎‘𝑥) = 0 )
139	138	oveq1d 7408	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → ((𝑎‘𝑥) · 𝑥) = ( 0 · 𝑥))
140	5	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑅 ∈ Ring)
141	9	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝐼 ⊆ 𝐵)
142		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 )))
143	142	eldifad 3956	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥 ∈ 𝐼)
144	141, 143	sseldd 3979	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → 𝑥 ∈ 𝐵)
145	140, 144, 99	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → ( 0 · 𝑥) = 0 )
146	139, 145	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝑥 ∈ (𝐼 ∖ (𝑎 supp 0 ))) → ((𝑎‘𝑥) · 𝑥) = 0 )
147	146, 70	suppss2 8167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)) supp 0 ) ⊆ (𝑎 supp 0 ))
148	87, 147	ssfid 9250	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)) supp 0 ) ∈ Fin)
149	2, 3, 69, 70, 134, 148	gsumcl2 19741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) ∈ 𝐵)
150	2, 67	cmncom 19630	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CMnd ∧ (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) ∈ 𝐵 ∧ ((𝑎‘𝑋) · 𝑋) ∈ 𝐵) → ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) + ((𝑎‘𝑋) · 𝑋)) = (((𝑎‘𝑋) · 𝑋) + (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)))))
151	69, 149, 120, 150	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥))) + ((𝑎‘𝑋) · 𝑋)) = (((𝑎‘𝑋) · 𝑋) + (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)))))
152	110, 125, 151	3eqtrd 2775	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) → (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))) = (((𝑎‘𝑋) · 𝑋) + (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)))))
153	152	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))) = (((𝑎‘𝑋) · 𝑋) + (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)))))
154	66, 153	eqtrd 2771	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → 𝐴 = (((𝑎‘𝑋) · 𝑋) + (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ ((𝑎‘𝑥) · 𝑥)))))
155	17, 19, 26, 65, 154	2rspcedvdw 3621	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ 𝑎 finSupp 0 ) ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) → ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖))
156	155	anasss 467	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))) ∧ (𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))))) → ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖))
157	156	r19.29an 3157	. . 3 ⊢ ((𝜑 ∧ ∃𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))))) → ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖))
158	27	a1i 11	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐵 ∈ V)
159	6	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐼 ∈ (LIdeal‘𝑅))
160	71	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → {𝑋} ∈ V)
161	159, 160	unexd 7724	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∪ {𝑋}) ∈ V)
162		simp-4r 782	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → 𝑟 ∈ 𝐵)
163		eqid 2731	. . . . . . . . . . . 12 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
164	2, 163	ringidcl 20040	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ 𝐵)
165	5, 164	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (1_r‘𝑅) ∈ 𝐵)
166	165	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → (1_r‘𝑅) ∈ 𝐵)
167	162, 166	ifcld 4568	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)) ∈ 𝐵)
168	82	ad4antr 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → 0 ∈ 𝐵)
169	167, 168	ifcld 4568	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ (𝐼 ∪ {𝑋})) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) ∈ 𝐵)
170	169	fmpttd 7099	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )):(𝐼 ∪ {𝑋})⟶𝐵)
171	158, 161, 170	elmapdd 8818	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋})))
172		breq1 5144	. . . . . . 7 ⊢ (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) → (𝑎 finSupp 0 ↔ (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) finSupp 0 ))
173		fveq1 6877	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) → (𝑎‘𝑥) = ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥))
174	173	oveq1d 7408	. . . . . . . . . 10 ⊢ (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) → ((𝑎‘𝑥) · 𝑥) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))
175	174	mpteq2dv 5243	. . . . . . . . 9 ⊢ (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)) = (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))
176	175	oveq2d 7409	. . . . . . . 8 ⊢ (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) → (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))) = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))))
177	176	eqeq2d 2742	. . . . . . 7 ⊢ (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) → (𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥))) ↔ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))))
178	172, 177	anbi12d 631	. . . . . 6 ⊢ (𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) → ((𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) ↔ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))))))
179	178	adantl 482	. . . . 5 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑎 = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))) → ((𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) ↔ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))))))
180		eqid 2731	. . . . . . 7 ⊢ (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) = (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))
181		prfi 9305	. . . . . . . 8 ⊢ {𝑋, 𝑖} ∈ Fin
182	181	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → {𝑋, 𝑖} ∈ Fin)
183		simp-4r 782	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ {𝑋, 𝑖}) → 𝑟 ∈ 𝐵)
184	165	ad4antr 730	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ {𝑋, 𝑖}) → (1_r‘𝑅) ∈ 𝐵)
185	183, 184	ifcld 4568	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑦 ∈ {𝑋, 𝑖}) → if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)) ∈ 𝐵)
186	82	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 0 ∈ 𝐵)
187	180, 161, 182, 185, 186	mptiffisupp 31786	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) finSupp 0 )
188	68	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ CMnd)
189	159, 8	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐼 ⊆ 𝐵)
190		simplr 767	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖 ∈ 𝐼)
191	189, 190	sseldd 3979	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖 ∈ 𝐵)
192	5	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ Ring)
193		simpllr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑟 ∈ 𝐵)
194	11	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑋 ∈ 𝐵)
195	2, 4, 192, 193, 194	ringcld 20037	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑟 · 𝑋) ∈ 𝐵)
196	2, 67	cmncom 19630	. . . . . . . . 9 ⊢ ((𝑅 ∈ CMnd ∧ 𝑖 ∈ 𝐵 ∧ (𝑟 · 𝑋) ∈ 𝐵) → (𝑖 + (𝑟 · 𝑋)) = ((𝑟 · 𝑋) + 𝑖))
197	188, 191, 195, 196	syl3anc 1371	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑖 + (𝑟 · 𝑋)) = ((𝑟 · 𝑋) + 𝑖))
198	188	cmnmndd 19636	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑅 ∈ Mnd)
199		eqid 2731	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))
200	191, 2	eleqtrdi 2842	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖 ∈ (Base‘𝑅))
201	3, 198, 159, 190, 199, 200	gsummptif1n0 19793	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))) = 𝑖)
202		fveq2 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑖 → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) = ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑖))
203		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑖 → 𝑥 = 𝑖)
204	202, 203	oveq12d 7411	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑖 → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑖) · 𝑖))
205	204	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑖) · 𝑖))
206		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑦 = 𝑖)
207		prid2g 4758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ 𝐼 → 𝑖 ∈ {𝑋, 𝑖})
208	207	ad5antlr 733	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑖 ∈ {𝑋, 𝑖})
209	206, 208	eqeltrd 2832	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑦 ∈ {𝑋, 𝑖})
210	209	iftrued 4530	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) = if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)))
211	190	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → 𝑖 ∈ 𝐼)
212	211	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑖 ∈ 𝐼)
213	206, 212	eqeltrd 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → 𝑦 ∈ 𝐼)
214	105	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ¬ 𝑋 ∈ 𝐼)
215	214	ad3antrrr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → ¬ 𝑋 ∈ 𝐼)
216		nelneq 2856	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝐼) → ¬ 𝑦 = 𝑋)
217	213, 215, 216	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → ¬ 𝑦 = 𝑋)
218	217	iffalsed 4533	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)) = (1_r‘𝑅))
219	210, 218	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) ∧ 𝑦 = 𝑖) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) = (1_r‘𝑅))
220	30, 211	sselid 3976	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → 𝑖 ∈ (𝐼 ∪ {𝑋}))
221	192	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → 𝑅 ∈ Ring)
222	221, 164	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → (1_r‘𝑅) ∈ 𝐵)
223	180, 219, 220, 222	fvmptd2 6992	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑖) = (1_r‘𝑅))
224	223	oveq1d 7408	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑖) · 𝑖) = ((1_r‘𝑅) · 𝑖))
225	191	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → 𝑖 ∈ 𝐵)
226	2, 4, 163, 221, 225	ringlidmd 20046	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → ((1_r‘𝑅) · 𝑖) = 𝑖)
227	205, 224, 226	3eqtrrd 2776	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ 𝑥 = 𝑖) → 𝑖 = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))
228		eleq1w 2815	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ {𝑋, 𝑖} ↔ 𝑥 ∈ {𝑋, 𝑖}))
229		eqeq1 2735	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑋 ↔ 𝑥 = 𝑋))
230	229	ifbid 4545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑥 → if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)) = if(𝑥 = 𝑋, 𝑟, (1_r‘𝑅)))
231	228, 230	ifbieq1d 4546	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) = if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))
232		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥 ∈ 𝐼)
233	30, 232	sselid 3976	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
234	193	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑟 ∈ 𝐵)
235	165	ad5antr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → (1_r‘𝑅) ∈ 𝐵)
236	234, 235	ifcld 4568	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → if(𝑥 = 𝑋, 𝑟, (1_r‘𝑅)) ∈ 𝐵)
237	186	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 0 ∈ 𝐵)
238	236, 237	ifcld 4568	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) ∈ 𝐵)
239	180, 231, 233, 238	fvmptd3 7007	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) = if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))
240	214	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → ¬ 𝑋 ∈ 𝐼)
241		nelne2 3039	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ 𝐼 ∧ ¬ 𝑋 ∈ 𝐼) → 𝑥 ≠ 𝑋)
242	232, 240, 241	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥 ≠ 𝑋)
243		neqne 2947	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑥 = 𝑖 → 𝑥 ≠ 𝑖)
244	243	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥 ≠ 𝑖)
245	242, 244	nelprd 4653	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → ¬ 𝑥 ∈ {𝑋, 𝑖})
246	245	iffalsed 4533	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → if(𝑥 ∈ {𝑋, 𝑖}, if(𝑥 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) = 0 )
247	239, 246	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) = 0 )
248	247	oveq1d 7408	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥) = ( 0 · 𝑥))
249	192	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑅 ∈ Ring)
250	189	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝐼 ⊆ 𝐵)
251	250, 232	sseldd 3979	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 𝑥 ∈ 𝐵)
252	249, 251, 99	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → ( 0 · 𝑥) = 0 )
253	248, 252	eqtr2d 2772	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) ∧ ¬ 𝑥 = 𝑖) → 0 = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))
254	227, 253	ifeqda 4558	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ 𝐼) → if(𝑥 = 𝑖, 𝑖, 0 ) = (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))
255	254	mpteq2dva 5241	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 )) = (𝑥 ∈ 𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))
256	255	oveq2d 7409	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑖, 𝑖, 0 ))) = (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))))
257	201, 256	eqtr3d 2773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝑖 = (𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))))
258		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
259		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑥 = 𝑋)
260	194	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑋 ∈ 𝐵)
261		prid1g 4757	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ {𝑋, 𝑖})
262	260, 261	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑋 ∈ {𝑋, 𝑖})
263	259, 262	eqeltrd 2832	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑥 ∈ {𝑋, 𝑖})
264	258, 263	eqeltrd 2832	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 ∈ {𝑋, 𝑖})
265	264	iftrued 4530	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) = if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)))
266	258, 259	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑋)
267	266	iftrued 4530	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)) = 𝑟)
268	265, 267	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) ∧ 𝑦 = 𝑥) → if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ) = 𝑟)
269		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
270	116	ad4antr 730	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑋 ∈ {𝑋})
271	270, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑋 ∈ (𝐼 ∪ {𝑋}))
272	269, 271	eqeltrd 2832	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
273	193	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → 𝑟 ∈ 𝐵)
274	180, 268, 272, 273	fvmptd2 6992	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) = 𝑟)
275	274, 269	oveq12d 7411	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 = 𝑋) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥) = (𝑟 · 𝑋))
276	2, 198, 194, 195, 275	gsumsnd 19779	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))) = (𝑟 · 𝑋))
277	276	eqcomd 2737	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑟 · 𝑋) = (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))))
278	257, 277	oveq12d 7411	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑖 + (𝑟 · 𝑋)) = ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))) + (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))))
279	197, 278	eqtr3d 2773	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑟 · 𝑋) + 𝑖) = ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))) + (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))))
280		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐴 = ((𝑟 · 𝑋) + 𝑖))
281	5	ad4antr 730	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑅 ∈ Ring)
282	170	ffvelcdmda 7071	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) ∈ 𝐵)
283	13	ad4antr 730	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
284		simpr 485	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
285	283, 284	sseldd 3979	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → 𝑥 ∈ 𝐵)
286	2, 4, 281, 282, 285	ringcld 20037	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ (𝐼 ∪ {𝑋})) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥) ∈ 𝐵)
287	161	mptexd 7210	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)) ∈ V)
288		funmpt 6575	. . . . . . . . . 10 ⊢ Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))
289	288	a1i 11	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → Fun (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))
290	187	fsuppimpd 9352	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ) ∈ Fin)
291		nfv 1917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦(((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖))
292	291, 169, 180	fnmptd 6678	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) Fn (𝐼 ∪ {𝑋}))
293	292	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → (𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) Fn (𝐼 ∪ {𝑋}))
294	161	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → (𝐼 ∪ {𝑋}) ∈ V)
295	186	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → 0 ∈ 𝐵)
296		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 )))
297	293, 294, 295, 296	fvdifsupp 31777	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) = 0 )
298	297	oveq1d 7408	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥) = ( 0 · 𝑥))
299	5	ad4antr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → 𝑅 ∈ Ring)
300	13	ad4antr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → (𝐼 ∪ {𝑋}) ⊆ 𝐵)
301	296	eldifad 3956	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → 𝑥 ∈ (𝐼 ∪ {𝑋}))
302	300, 301	sseldd 3979	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → 𝑥 ∈ 𝐵)
303	299, 302, 99	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → ( 0 · 𝑥) = 0 )
304	298, 303	eqtrd 2771	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) ∧ 𝑥 ∈ ((𝐼 ∪ {𝑋}) ∖ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))) → (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥) = 0 )
305	304, 161	suppss2 8167	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)) supp 0 ) ⊆ ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) supp 0 ))
306	290, 305	ssfid 9250	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)) supp 0 ) ∈ Fin)
307	287, 186, 289, 306	isfsuppd 9349	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)) finSupp 0 )
308	214, 107	sylibr 233	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∩ {𝑋}) = ∅)
309		eqidd 2732	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝐼 ∪ {𝑋}) = (𝐼 ∪ {𝑋}))
310	2, 3, 67, 188, 161, 286, 307, 308, 309	gsumsplit2 19756	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))) = ((𝑅 Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))) + (𝑅 Σ_g (𝑥 ∈ {𝑋} ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))))
311	279, 280, 310	3eqtr4d 2781	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥))))
312	187, 311	jca 512	. . . . 5 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 )) finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ (((𝑦 ∈ (𝐼 ∪ {𝑋}) ↦ if(𝑦 ∈ {𝑋, 𝑖}, if(𝑦 = 𝑋, 𝑟, (1_r‘𝑅)), 0 ))‘𝑥) · 𝑥)))))
313	171, 179, 312	rspcedvd 3611	. . . 4 ⊢ ((((𝜑 ∧ 𝑟 ∈ 𝐵) ∧ 𝑖 ∈ 𝐼) ∧ 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ∃𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))))
314	313	r19.29ffa 31576	. . 3 ⊢ ((𝜑 ∧ ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)) → ∃𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))))
315	157, 314	impbida 799	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝐵 ↑_m (𝐼 ∪ {𝑋}))(𝑎 finSupp 0 ∧ 𝐴 = (𝑅 Σ_g (𝑥 ∈ (𝐼 ∪ {𝑋}) ↦ ((𝑎‘𝑥) · 𝑥)))) ↔ ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))
316	14, 315	bitrd 278	1 ⊢ (𝜑 → (𝐴 ∈ (𝑁‘(𝐼 ∪ {𝑋})) ↔ ∃𝑟 ∈ 𝐵 ∃𝑖 ∈ 𝐼 𝐴 = ((𝑟 · 𝑋) + 𝑖)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∃wrex 3069 Vcvv 3473 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∅c0 4318 ifcif 4522 {csn 4622 {cpr 4624 class class class wbr 5141 ↦ cmpt 5224 ↾ cres 5671 Fun wfun 6526 Fn wfn 6527 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 supp csupp 8128 ↑_m cmap 8803 Fincfn 8922 finSupp cfsupp 9344 Basecbs 17126 +_gcplusg 17179 ._rcmulr 17180 0_gc0g 17367 Σ_g cgsu 17368 CMndccmn 19612 1_rcur 19963 Ringcrg 20014 LIdealclidl 20732 RSpancrsp 20733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-sup 9419 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-fzo 13610 df-seq 13949 df-hash 14273 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17369 df-gsum 17370 df-prds 17375 df-pws 17377 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mulg 18923 df-subg 18975 df-ghm 19056 df-cntz 19147 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-nzr 20242 df-subrg 20310 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lmhm 20582 df-lbs 20635 df-sra 20734 df-rgmod 20735 df-lidl 20736 df-rsp 20737 df-dsmm 21220 df-frlm 21235 df-uvc 21271

This theorem is referenced by: qsdrngilem 32452

W3C validator