Theorem psrass1 21901

Description: Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)

Hypotheses

Ref	Expression
psrring.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrring.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrring.r	⊢ (𝜑 → 𝑅 ∈ Ring)
psrass.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
psrass.t	⊢ × = (.r‘𝑆)
psrass.b	⊢ 𝐵 = (Base‘𝑆)
psrass.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
psrass.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
psrass.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
psrass1	⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Distinct variable groups:   𝑓,𝐼   𝑅,𝑓   𝑓,𝑋   𝑓,𝑍   𝑓,𝑌

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐷(𝑓)   𝑆(𝑓)   × (𝑓)   𝑉(𝑓)

Proof of Theorem psrass1

Dummy variables 𝑥 𝑘 𝑧 𝑔 ℎ 𝑗 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrring.s	. . . 4 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		eqid 2731	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		psrass.d	. . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
4		psrass.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
5		psrass.t	. . . . 5 ⊢ × = (.r‘𝑆)
6		psrring.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		psrass.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		psrass.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 4, 5, 6, 7, 8	psrmulcl 21883	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10		psrass.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11	1, 4, 5, 6, 9, 10	psrmulcl 21883	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
12	1, 2, 3, 4, 11	psrelbas 21871	. . 3 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
13	12	ffnd 6652	. 2 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
14	1, 4, 5, 6, 8, 10	psrmulcl 21883	. . . . 5 ⊢ (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
15	1, 4, 5, 6, 7, 14	psrmulcl 21883	. . . 4 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
16	1, 2, 3, 4, 15	psrelbas 21871	. . 3 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
17	16	ffnd 6652	. 2 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18		eqid 2731	. . . . 5 ⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}
19		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
20	6	ringcmnd 20202	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
21	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ CMnd)
22		eqid 2731	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
23	6	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑅 ∈ Ring)
24	1, 2, 3, 4, 7	psrelbas 21871	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
25	24	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
26		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥})
27		breq1 5092	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑗 ∘r ≤ 𝑥))
28	27	elrab 3642	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥))
29	26, 28	sylib 218	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥))
30	29	simpld 494	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∈ 𝐷)
31	25, 30	ffvelcdmd 7018	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
32	31	adantr 480	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
33	1, 2, 3, 4, 8	psrelbas 21871	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
34	33	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
35		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})
36		breq1 5092	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑛 → (ℎ ∘r ≤ (𝑥 ∘f − 𝑗) ↔ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
37	36	elrab 3642	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↔ (𝑛 ∈ 𝐷 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
38	35, 37	sylib 218	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑛 ∈ 𝐷 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
39	38	simpld 494	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∈ 𝐷)
40	34, 39	ffvelcdmd 7018	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑌‘𝑛) ∈ (Base‘𝑅))
41	1, 2, 3, 4, 10	psrelbas 21871	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍:𝐷⟶(Base‘𝑅))
42	41	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
43		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷)
44	3	psrbagf 21855	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐷 → 𝑗:𝐼⟶ℕ0)
45	30, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0)
46	29	simprd 495	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∘r ≤ 𝑥)
47	3	psrbagcon 21862	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥))
48	43, 45, 46, 47	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥))
49	48	simpld 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑗) ∈ 𝐷)
50	49	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑥 ∘f − 𝑗) ∈ 𝐷)
51	3	psrbagf 21855	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐷 → 𝑛:𝐼⟶ℕ0)
52	39, 51	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛:𝐼⟶ℕ0)
53	38	simprd 495	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗))
54	3	psrbagcon 21862	. . . . . . . . . . 11 ⊢ (((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ 𝑛:𝐼⟶ℕ0 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)) → (((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∈ 𝐷 ∧ ((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∘r ≤ (𝑥 ∘f − 𝑗)))
55	50, 52, 53, 54	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∈ 𝐷 ∧ ((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∘r ≤ (𝑥 ∘f − 𝑗)))
56	55	simpld 494	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∈ 𝐷)
57	42, 56	ffvelcdmd 7018	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)) ∈ (Base‘𝑅))
58	2, 22, 23, 40, 57	ringcld 20178	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))) ∈ (Base‘𝑅))
59	2, 22, 23, 32, 58	ringcld 20178	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ (Base‘𝑅))
60	59	anasss 466	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ (Base‘𝑅))
61		fveq2 6822	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑌‘𝑛) = (𝑌‘(𝑘 ∘f − 𝑗)))
62		oveq2 7354	. . . . . . . 8 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑥 ∘f − 𝑗) ∘f − 𝑛) = ((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))
63	62	fveq2d 6826	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)) = (𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))
64	61, 63	oveq12d 7364	. . . . . 6 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))) = ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))))
65	64	oveq2d 7362	. . . . 5 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))
66	3, 18, 19, 2, 21, 60, 65	psrass1lem 21869	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))))
67	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋 ∈ 𝐵)
68	8	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵)
69		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥})
70		breq1 5092	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑘 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑘 ∘r ≤ 𝑥))
71	70	elrab 3642	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥))
72	69, 71	sylib 218	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥))
73	72	simpld 494	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∈ 𝐷)
74	1, 4, 22, 5, 3, 67, 68, 73	psrmulval 21881	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))))))
75	74	oveq1d 7361	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))
76		eqid 2731	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
77	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring)
78	3	psrbaglefi 21863	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin)
79	73, 78	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin)
80	41	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
81		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷)
82	3	psrbagf 21855	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐷 → 𝑘:𝐼⟶ℕ0)
83	73, 82	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0)
84	72	simprd 495	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∘r ≤ 𝑥)
85	3	psrbagcon 21862	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑘:𝐼⟶ℕ0 ∧ 𝑘 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥))
86	81, 83, 84, 85	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥))
87	86	simpld 494	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑘) ∈ 𝐷)
88	80, 87	ffvelcdmd 7018	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))
89	6	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑅 ∈ Ring)
90	24	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
91		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘})
92		breq1 5092	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑗 → (ℎ ∘r ≤ 𝑘 ↔ 𝑗 ∘r ≤ 𝑘))
93	92	elrab 3642	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘))
94	91, 93	sylib 218	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘))
95	94	simpld 494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∈ 𝐷)
96	90, 95	ffvelcdmd 7018	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
97	33	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
98	73	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷)
99	95, 44	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗:𝐼⟶ℕ0)
100	94	simprd 495	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∘r ≤ 𝑘)
101	3	psrbagcon 21862	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑘) → ((𝑘 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘f − 𝑗) ∘r ≤ 𝑘))
102	98, 99, 100, 101	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑘 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘f − 𝑗) ∘r ≤ 𝑘))
103	102	simpld 494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ 𝐷)
104	97, 103	ffvelcdmd 7018	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅))
105	2, 22, 89, 96, 104	ringcld 20178	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))) ∈ (Base‘𝑅))
106		eqid 2731	. . . . . . . . 9 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))))
107		fvex 6835	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
108	107	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (0g‘𝑅) ∈ V)
109	106, 79, 105, 108	fsuppmptdm 9260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))) finSupp (0g‘𝑅))
110	2, 76, 22, 77, 79, 88, 105, 109	gsummulc1 20234	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = ((𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))
111	88	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))
112	2, 22	ringass 20171	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((𝑋‘𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅) ∧ (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))))
113	89, 96, 104, 111, 112	syl13anc 1374	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))))
114	3	psrbagf 21855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐷 → 𝑥:𝐼⟶ℕ0)
115	114	ad3antlr 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥:𝐼⟶ℕ0)
116	115	ffvelcdmda 7017	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈ ℕ0)
117	83	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
118	117	ffvelcdmda 7017	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
119	99	ffvelcdmda 7017	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈ ℕ0)
120		nn0cn 12391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ)
121		nn0cn 12391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
122		nn0cn 12391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ)
123		nnncan2 11398	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥‘𝑧) ∈ ℂ ∧ (𝑘‘𝑧) ∈ ℂ ∧ (𝑗‘𝑧) ∈ ℂ) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
124	120, 121, 122, 123	syl3an 1160	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥‘𝑧) ∈ ℕ0 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑗‘𝑧) ∈ ℕ0) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
125	116, 118, 119, 124	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
126	125	mpteq2dva 5182	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧))))
127		psrring.i	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
128	127	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑉)
129		ovexd 7381	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V)
130		ovexd 7381	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧) − (𝑗‘𝑧)) ∈ V)
131	115	feqmptd 6890	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧)))
132	99	feqmptd 6890	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
133	128, 116, 119, 131, 132	offval2 7630	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑥 ∘f − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑗‘𝑧))))
134	117	feqmptd 6890	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
135	128, 118, 119, 134, 132	offval2 7630	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧) − (𝑗‘𝑧))))
136	128, 129, 130, 133, 135	offval2 7630	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)) = (𝑧 ∈ 𝐼 ↦ (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧)))))
137	128, 116, 118, 131, 134	offval2 7630	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑥 ∘f − 𝑘) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑘‘𝑧))))
138	126, 136, 137	3eqtr4d 2776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)) = (𝑥 ∘f − 𝑘))
139	138	fveq2d 6826	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))) = (𝑍‘(𝑥 ∘f − 𝑘)))
140	139	oveq2d 7362	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))) = ((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))
141	140	oveq2d 7362	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))))
142	113, 141	eqtr4d 2769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))
143	142	mpteq2dva 5182	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) = (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))))))
144	143	oveq2d 7362	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ (((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗)))(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))
145	75, 110, 144	3eqtr2d 2772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))) = (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))
146	145	mpteq2dva 5182	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘)))) = (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))))))))
147	146	oveq2d 7362	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘(𝑘 ∘f − 𝑗))(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))))))))
148	8	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵)
149	10	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍 ∈ 𝐵)
150	1, 4, 22, 5, 3, 148, 149, 49	psrmulval 21881	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))))
151	150	oveq2d 7362	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))
152	6	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring)
153	3	psrbaglefi 21863	. . . . . . . . 9 ⊢ ((𝑥 ∘f − 𝑗) ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin)
154	49, 153	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin)
155		ovex 7379	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m 𝐼) ∈ V
156	3, 155	rab2ex 5278	. . . . . . . . . . . 12 ⊢ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ V
157	156	mptex 7157	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V
158		funmpt 6519	. . . . . . . . . . 11 ⊢ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))
159	157, 158, 107	3pm3.2i 1340	. . . . . . . . . 10 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∧ (0g‘𝑅) ∈ V)
160	159	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∧ (0g‘𝑅) ∈ V))
161		suppssdm 8107	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ dom (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))
162		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) = (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))
163	162	dmmptss 6188	. . . . . . . . . . 11 ⊢ dom (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}
164	161, 163	sstri 3939	. . . . . . . . . 10 ⊢ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}
165	164	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})
166		suppssfifsupp 9264	. . . . . . . . 9 ⊢ ((((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∧ (0g‘𝑅) ∈ V) ∧ ({ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin ∧ ((𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) supp (0g‘𝑅)) ⊆ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)})) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) finSupp (0g‘𝑅))
167	160, 154, 165, 166	syl12anc 836	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) finSupp (0g‘𝑅))
168	2, 76, 22, 152, 154, 31, 58, 167	gsummulc2 20235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))) = ((𝑋‘𝑗)(.r‘𝑅)(𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))
169	151, 168	eqtr4d 2769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))) = (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))
170	169	mpteq2dva 5182	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗)))) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))))))
171	170	oveq2d 7362	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (𝑅 Σg (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))))))))
172	66, 147, 171	3eqtr4d 2776	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))))))
173	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
174	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ∈ 𝐵)
175	1, 4, 22, 5, 3, 173, 174, 19	psrmulval 21881	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r‘𝑅)(𝑍‘(𝑥 ∘f − 𝑘))))))
176	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑋 ∈ 𝐵)
177	14	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
178	1, 4, 22, 5, 3, 176, 177, 19	psrmulval 21881	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↦ ((𝑋‘𝑗)(.r‘𝑅)((𝑌 × 𝑍)‘(𝑥 ∘f − 𝑗))))))
179	172, 175, 178	3eqtr4d 2776	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
180	13, 17, 179	eqfnfvd 6967	1 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ⊆ wss 3897   class class class wbr 5089   ↦ cmpt 5170  ^◡ccnv 5613  dom cdm 5614   “ cima 5617  Fun wfun 6475  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608   ∘_r cofr 7609   supp csupp 8090   ↑_m cmap 8750  Fincfn 8869   finSupp cfsupp 9245  ℂcc 11004   ≤ cle 11147   − cmin 11344  ℕcn 12125  ℕ₀cn0 12381  Basecbs 17120  ._rcmulr 17162  0_gc0g 17343   Σ_g cgsu 17344  CMndccmn 19692  Ringcrg 20151   mPwSer cmps 21841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-ofr 7611  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-pm 8753  df-ixp 8822  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-oi 9396  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-seq 13909  df-hash 14238  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-tset 17180  df-0g 17345  df-gsum 17346  df-mre 17488  df-mrc 17489  df-acs 17491  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-mhm 18691  df-submnd 18692  df-grp 18849  df-minusg 18850  df-mulg 18981  df-ghm 19125  df-cntz 19229  df-cmn 19694  df-abl 19695  df-mgp 20059  df-ur 20100  df-ring 20153  df-psr 21846

This theorem is referenced by:  psrring  21907