Theorem mnringmulrcld 44816

Description: Monoid rings are closed under multiplication. (Contributed by Rohan Ridenour, 14-May-2024.)

Hypotheses

Ref	Expression
mnringmulrcld.2	⊢ 𝐹 = (𝑅 MndRing 𝑀)
mnringmulrcld.3	⊢ 𝐵 = (Base‘𝐹)
mnringmulrcld.1	⊢ 𝐴 = (Base‘𝑀)
mnringmulrcld.4	⊢ · = (.r‘𝐹)
mnringmulrcld.5	⊢ (𝜑 → 𝑅 ∈ Ring)
mnringmulrcld.6	⊢ (𝜑 → 𝑀 ∈ 𝑈)
mnringmulrcld.7	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mnringmulrcld.8	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
mnringmulrcld	⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)

Proof of Theorem mnringmulrcld

Dummy variables 𝑎 𝑏 𝑝 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnringmulrcld.2	. . 3 ⊢ 𝐹 = (𝑅 MndRing 𝑀)
2		mnringmulrcld.3	. . 3 ⊢ 𝐵 = (Base‘𝐹)
3		eqid 2765	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
4		eqid 2765	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
5		mnringmulrcld.1	. . 3 ⊢ 𝐴 = (Base‘𝑀)
6		eqid 2765	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
7		mnringmulrcld.4	. . 3 ⊢ · = (.r‘𝐹)
8		mnringmulrcld.5	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		mnringmulrcld.6	. . 3 ⊢ (𝜑 → 𝑀 ∈ 𝑈)
10		mnringmulrcld.7	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		mnringmulrcld.8	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	mnringmulrvald 44815	. 2 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))))
13		eqid 2765	. . 3 ⊢ (0g‘𝐹) = (0g‘𝐹)
14	1, 8, 9	mnringlmodd 44814	. . . 4 ⊢ (𝜑 → 𝐹 ∈ LMod)
15		lmodcmn 21000	. . . 4 ⊢ (𝐹 ∈ LMod → 𝐹 ∈ CMnd)
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → 𝐹 ∈ CMnd)
17	5	fvexi 6885	. . . . 5 ⊢ 𝐴 ∈ V
18	17, 17	xpex 7740	. . . 4 ⊢ (𝐴 × 𝐴) ∈ V
19	18	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 × 𝐴) ∈ V)
20	8	3ad2ant1 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑅 ∈ Ring)
21		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘𝑅)
22	1, 2, 5, 21, 8, 9, 10	mnringbasefd 44806	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋:𝐴⟶(Base‘𝑅))
23	22	3ad2ant1 1149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑋:𝐴⟶(Base‘𝑅))
24		simp2 1153	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑎 ∈ 𝐴)
25	23, 24	ffvelcdmd 7070	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑋‘𝑎) ∈ (Base‘𝑅))
26	1, 2, 5, 21, 8, 9, 11	mnringbasefd 44806	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌:𝐴⟶(Base‘𝑅))
27	26	3ad2ant1 1149	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑌:𝐴⟶(Base‘𝑅))
28		simp3 1154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ 𝐴)
29	27, 28	ffvelcdmd 7070	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑌‘𝑏) ∈ (Base‘𝑅))
30	21, 3	ringcl 20323	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑎) ∈ (Base‘𝑅) ∧ (𝑌‘𝑏) ∈ (Base‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) ∈ (Base‘𝑅))
31	20, 25, 29, 30	syl3anc 1394	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) ∈ (Base‘𝑅))
32	21, 4	ring0cl 20341	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
33	20, 32	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (0g‘𝑅) ∈ (Base‘𝑅))
34	31, 33	ifcld 4530	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)) ∈ (Base‘𝑅))
35	34	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑖 ∈ 𝐴) → if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)) ∈ (Base‘𝑅))
36	35	fmpttd 7100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))):𝐴⟶(Base‘𝑅))
37	21	fvexi 6885	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
38	37, 17	elmap 8857	. . . . . . . . 9 ⊢ ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ↔ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))):𝐴⟶(Base‘𝑅))
39	36, 38	sylibr 237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴))
40	17	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝐴 ∈ V)
41		eqid 2765	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))
42	40, 33, 41	sniffsupp 9348	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) finSupp (0g‘𝑅))
43	39, 42	jca 520	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ∧ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) finSupp (0g‘𝑅)))
44	9	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → 𝑀 ∈ 𝑈)
45	1, 2, 5, 21, 4, 20, 44	mnringelbased 44805	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵 ↔ ((𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ ((Base‘𝑅) ↑m 𝐴) ∧ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) finSupp (0g‘𝑅))))
46	43, 45	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵)
47	46	3expb 1136	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵)
48	47	ralrimivva 3208	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵)
49		eqid 2765	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
50	49	fmpo 8053	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ 𝐵 ↔ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))):(𝐴 × 𝐴)⟶𝐵)
51	48, 50	sylib 221	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))):(𝐴 × 𝐴)⟶𝐵)
52	17, 17	mpoex 8064	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) ∈ V
53	52	a1i 11	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) ∈ V)
54	51	ffnd 6696	. . . 4 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) Fn (𝐴 × 𝐴))
55	13	fvexi 6885	. . . . 5 ⊢ (0g‘𝐹) ∈ V
56	55	a1i 11	. . . 4 ⊢ (𝜑 → (0g‘𝐹) ∈ V)
57	1, 2, 4, 8, 9, 10	mnringbasefsuppd 44807	. . . . . 6 ⊢ (𝜑 → 𝑋 finSupp (0g‘𝑅))
58	57	fsuppimpd 9317	. . . . 5 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) ∈ Fin)
59	1, 2, 4, 8, 9, 11	mnringbasefsuppd 44807	. . . . . 6 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅))
60	59	fsuppimpd 9317	. . . . 5 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin)
61		xpfi 9267	. . . . 5 ⊢ (((𝑋 supp (0g‘𝑅)) ∈ Fin ∧ (𝑌 supp (0g‘𝑅)) ∈ Fin) → ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∈ Fin)
62	58, 60, 61	syl2anc 595	. . . 4 ⊢ (𝜑 → ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∈ Fin)
63		elxpi 5674	. . . . . . 7 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)))
64		simpl 487	. . . . . . . 8 ⊢ ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑝 = ⟨𝑎, 𝑏⟩)
65	64	2eximi 1859	. . . . . . 7 ⊢ (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
66	63, 65	syl 18	. . . . . 6 ⊢ (𝑝 ∈ (𝐴 × 𝐴) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
67	66	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → ∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩)
68		nfv 1937	. . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴))
69		nfv 1937	. . . . . . 7 ⊢ Ⅎ𝑎 𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))
70		nfmpo1 7480	. . . . . . . . 9 ⊢ Ⅎ𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
71		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑎𝑝
72	70, 71	nffv 6881	. . . . . . . 8 ⊢ Ⅎ𝑎((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝)
73		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑎(0g‘𝐹)
74	72, 73	nfeq 2940	. . . . . . 7 ⊢ Ⅎ𝑎((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)
75	69, 74	nfor 1927	. . . . . 6 ⊢ Ⅎ𝑎(𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))
76		nfv 1937	. . . . . . 7 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴))
77		nfv 1937	. . . . . . . 8 ⊢ Ⅎ𝑏 𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))
78		nfmpo2 7481	. . . . . . . . . 10 ⊢ Ⅎ𝑏(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
79		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑏𝑝
80	78, 79	nffv 6881	. . . . . . . . 9 ⊢ Ⅎ𝑏((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝)
81		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑏(0g‘𝐹)
82	80, 81	nfeq 2940	. . . . . . . 8 ⊢ Ⅎ𝑏((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)
83	77, 82	nfor 1927	. . . . . . 7 ⊢ Ⅎ𝑏(𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))
84		simp3 1154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
85		simp2 1153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 ∈ (𝐴 × 𝐴))
86	84, 85	eqeltrrd 2866	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴))
87		opelxp 5688	. . . . . . . . . 10 ⊢ (⟨𝑎, 𝑏⟩ ∈ (𝐴 × 𝐴) ↔ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
88	86, 87	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴))
89		ianor 997	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ↔ (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
90	22	ffnd 6696	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑋 Fn 𝐴)
91	17	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐴 ∈ V)
92	4	fvexi 6885	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0g‘𝑅) ∈ V
93	92	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
94		elsuppfn 8154	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋 Fn 𝐴 ∧ 𝐴 ∈ V ∧ (0g‘𝑅) ∈ V) → (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅))))
95	90, 91, 93, 94	syl3anc 1394	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ↔ (𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅))))
96	95	biimprd 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅)) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
97	96	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑎 ∈ 𝐴 ∧ (𝑋‘𝑎) ≠ (0g‘𝑅)) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
98	24, 97	mpand 707	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎) ≠ (0g‘𝑅) → 𝑎 ∈ (𝑋 supp (0g‘𝑅))))
99	98	necon1bd 2978	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) → (𝑋‘𝑎) = (0g‘𝑅)))
100	26	ffnd 6696	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑌 Fn 𝐴)
101		elsuppfn 8154	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑌 Fn 𝐴 ∧ 𝐴 ∈ V ∧ (0g‘𝑅) ∈ V) → (𝑏 ∈ (𝑌 supp (0g‘𝑅)) ↔ (𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅))))
102	100, 91, 93, 101	syl3anc 1394	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑏 ∈ (𝑌 supp (0g‘𝑅)) ↔ (𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅))))
103	102	biimprd 251	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅)) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
104	103	3ad2ant1 1149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑏 ∈ 𝐴 ∧ (𝑌‘𝑏) ≠ (0g‘𝑅)) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
105	28, 104	mpand 707	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑌‘𝑏) ≠ (0g‘𝑅) → 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
106	105	necon1bd 2978	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅)) → (𝑌‘𝑏) = (0g‘𝑅)))
107	99, 106	orim12d 979	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) → ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))))
108	107	imp 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (¬ 𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∨ ¬ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))) → ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅)))
109	89, 108	sylan2b 605	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))) → ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅)))
110		oveq1 7407	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋‘𝑎) = (0g‘𝑅) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)))
111	21, 3, 4	ringlz 20367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝑌‘𝑏) ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
112	20, 29, 111	syl2anc 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((0g‘𝑅)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
113	110, 112	sylan9eqr 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (𝑋‘𝑎) = (0g‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
114		oveq2 7408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑌‘𝑏) = (0g‘𝑅) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)))
115	21, 3, 4	ringrz 20368	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑎) ∈ (Base‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
116	20, 25, 115	syl2anc 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑋‘𝑎)(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
117	114, 116	sylan9eqr 2822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ (𝑌‘𝑏) = (0g‘𝑅)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
118	113, 117	jaodan 972	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
119	118	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) ∧ 𝑖 = (𝑎(+g‘𝑀)𝑏)) → ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)) = (0g‘𝑅))
120		eqidd 2766	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) ∧ ¬ 𝑖 = (𝑎(+g‘𝑀)𝑏)) → (0g‘𝑅) = (0g‘𝑅))
121	119, 120	ifeqda 4520	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)) = (0g‘𝑅))
122	121	mpteq2dv 5199	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)))
123		fconstmpt 5714	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 × {(0g‘𝑅)}) = (𝑖 ∈ 𝐴 ↦ (0g‘𝑅))
124	1, 4, 5, 8, 9	mnring0g2d 44810	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐴 × {(0g‘𝑅)}) = (0g‘𝐹))
125	123, 124	eqtr3id 2814	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
126	125	3ad2ant1 1149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
127	126	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ (0g‘𝑅)) = (0g‘𝐹))
128	122, 127	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ((𝑋‘𝑎) = (0g‘𝑅) ∨ (𝑌‘𝑏) = (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹))
129	109, 128	syldan 602	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ ¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹))
130	129	ex 417	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (¬ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
131	130	orrd 876	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
132	131	3expb 1136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
133	132	3adant3 1148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
134		eleq1 2853	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ ⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅)))))
135		opelxp 5688	. . . . . . . . . . . . 13 ⊢ (⟨𝑎, 𝑏⟩ ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))))
136	134, 135	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))))
137	136	3ad2ant3 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ↔ (𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅)))))
138		simp2l 1216	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑎 ∈ 𝐴)
139		simp2r 1217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑏 ∈ 𝐴)
140		eqidd 2766	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))))
141		simp3 1154	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → 𝑝 = ⟨𝑎, 𝑏⟩)
142	17	mptex 7211	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ V
143	142	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) ∈ V)
144	140, 141, 143	fvmpopr2d 7562	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) → ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
145	138, 139, 144	mpd3an23 1487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))
146	145	eqeq1d 2767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹) ↔ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹)))
147	137, 146	orbi12d 931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → ((𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)) ↔ ((𝑎 ∈ (𝑋 supp (0g‘𝑅)) ∧ 𝑏 ∈ (𝑌 supp (0g‘𝑅))) ∨ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))) = (0g‘𝐹))))
148	133, 147	mpbird 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)))
149	88, 148	syld3an2 1434	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴) ∧ 𝑝 = ⟨𝑎, 𝑏⟩) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)))
150	149	3expia 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))))
151	76, 83, 150	exlimd 2256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))))
152	68, 75, 151	exlimd 2256	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (∃𝑎∃𝑏 𝑝 = ⟨𝑎, 𝑏⟩ → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹))))
153	67, 152	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝑝 ∈ ((𝑋 supp (0g‘𝑅)) × (𝑌 supp (0g‘𝑅))) ∨ ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))‘𝑝) = (0g‘𝐹)))
154	53, 54, 56, 62, 153	finnzfsuppd 9321	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅)))) finSupp (0g‘𝐹))
155	2, 13, 16, 19, 51, 154	gsumcl 19976	. 2 ⊢ (𝜑 → (𝐹 Σg (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐴 ↦ (𝑖 ∈ 𝐴 ↦ if(𝑖 = (𝑎(+g‘𝑀)𝑏), ((𝑋‘𝑎)(.r‘𝑅)(𝑌‘𝑏)), (0g‘𝑅))))) ∈ 𝐵)
156	12, 155	eqeltrd 2865	1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  Vcvv 3457  ifcif 4483  {csn 4585  ⟨cop 4591   class class class wbr 5105   ↦ cmpt 5186   × cxp 5650   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   supp csupp 8144   ↑_m cmap 8812  Fincfn 8931   finSupp cfsupp 9309  Basecbs 17259  +_gcplusg 17300  ._rcmulr 17301  0_gc0g 17482   Σ_g cgsu 17483  CMndccmn 19841  Ringcrg 20306  LModclmod 20950   MndRing cmnring 44799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-hom 17324  df-cco 17325  df-0g 17484  df-gsum 17485  df-prds 17490  df-pws 17492  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-grp 18993  df-minusg 18994  df-sbg 18995  df-subg 19180  df-cntz 19378  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-subrg 20646  df-lmod 20952  df-lss 21022  df-sra 21263  df-rgmod 21264  df-dsmm 21842  df-frlm 21857  df-mnring 44800

This theorem is referenced by: (None)