Theorem psrass1 21174

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 2738	. . . 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 21157	. . . . 5 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10		psrass.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11	1, 4, 5, 6, 9, 10	psrmulcl 21157	. . . 4 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
12	1, 2, 3, 4, 11	psrelbas 21148	. . 3 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
13	12	ffnd 6601	. 2 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
14	1, 4, 5, 6, 8, 10	psrmulcl 21157	. . . . 5 ⊢ (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
15	1, 4, 5, 6, 7, 14	psrmulcl 21157	. . . 4 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
16	1, 2, 3, 4, 15	psrelbas 21148	. . 3 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
17	16	ffnd 6601	. 2 ⊢ (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18		eqid 2738	. . . . 5 ⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}
19		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
20		ringcmn 19820	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
21	6, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
22	21	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ CMnd)
23	6	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring)
24	23	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑅 ∈ Ring)
25	1, 2, 3, 4, 7	psrelbas 21148	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
26	25	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
27		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥})
28		breq1 5077	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑗 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑗 ∘r ≤ 𝑥))
29	28	elrab 3624	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥))
30	27, 29	sylib 217	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑥))
31	30	simpld 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∈ 𝐷)
32	26, 31	ffvelrnd 6962	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
33	32	adantr 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
34	1, 2, 3, 4, 8	psrelbas 21148	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
35	34	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
36		simpr 485	. . . . . . . . . . 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)})
37		breq1 5077	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑛 → (ℎ ∘r ≤ (𝑥 ∘f − 𝑗) ↔ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
38	37	elrab 3624	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↔ (𝑛 ∈ 𝐷 ∧ 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗)))
39	36, 38	sylib 217	. . . . . . . . . 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)))
40	39	simpld 495	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∈ 𝐷)
41	35, 40	ffvelrnd 6962	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑌‘𝑛) ∈ (Base‘𝑅))
42	1, 2, 3, 4, 10	psrelbas 21148	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍:𝐷⟶(Base‘𝑅))
43	42	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
44		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷)
45	3	psrbagf 21121	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ 𝐷 → 𝑗:𝐼⟶ℕ0)
46	31, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0)
47	30	simprd 496	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑗 ∘r ≤ 𝑥)
48	3	psrbagcon 21133	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥))
49	44, 46, 47, 48	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑗) ∘r ≤ 𝑥))
50	49	simpld 495	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑗) ∈ 𝐷)
51	50	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → (𝑥 ∘f − 𝑗) ∈ 𝐷)
52	3	psrbagf 21121	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐷 → 𝑛:𝐼⟶ℕ0)
53	40, 52	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛:𝐼⟶ℕ0)
54	39	simprd 496	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → 𝑛 ∘r ≤ (𝑥 ∘f − 𝑗))
55	3	psrbagcon 21133	. . . . . . . . . . 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)))
56	51, 53, 54, 55	syl3anc 1370	. . . . . . . . . 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)))
57	56	simpld 495	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)}) → ((𝑥 ∘f − 𝑗) ∘f − 𝑛) ∈ 𝐷)
58	43, 57	ffvelrnd 6962	. . . . . . . 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))
59		eqid 2738	. . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅)
60	2, 59	ringcl 19800	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑌‘𝑛) ∈ (Base‘𝑅) ∧ (𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)) ∈ (Base‘𝑅)) → ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))) ∈ (Base‘𝑅))
61	24, 41, 58, 60	syl3anc 1370	. . . . . . 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))
62	2, 59	ringcl 19800	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑗) ∈ (Base‘𝑅) ∧ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))) ∈ (Base‘𝑅)) → ((𝑋‘𝑗)(.r‘𝑅)((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ (Base‘𝑅))
63	24, 33, 61, 62	syl3anc 1370	. . . . . 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))
64	63	anasss 467	. . . . 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))
65		fveq2 6774	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑌‘𝑛) = (𝑌‘(𝑘 ∘f − 𝑗)))
66		oveq2 7283	. . . . . . . 8 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → ((𝑥 ∘f − 𝑗) ∘f − 𝑛) = ((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗)))
67	66	fveq2d 6778	. . . . . . 7 ⊢ (𝑛 = (𝑘 ∘f − 𝑗) → (𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)) = (𝑍‘((𝑥 ∘f − 𝑗) ∘f − (𝑘 ∘f − 𝑗))))
68	65, 67	oveq12d 7293	. . . . . 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)))))
69	68	oveq2d 7291	. . . . 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))))))
70	3, 18, 19, 2, 22, 64, 69	psrass1lem 21146	. . . 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)))))))))
71	7	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑋 ∈ 𝐵)
72	8	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵)
73		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥})
74		breq1 5077	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑘 → (𝑔 ∘r ≤ 𝑥 ↔ 𝑘 ∘r ≤ 𝑥))
75	74	elrab 3624	. . . . . . . . . . 11 ⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥} ↔ (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥))
76	73, 75	sylib 217	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑘 ∈ 𝐷 ∧ 𝑘 ∘r ≤ 𝑥))
77	76	simpld 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∈ 𝐷)
78	1, 4, 59, 5, 3, 71, 72, 77	psrmulval 21155	. . . . . . . 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))))))
79	78	oveq1d 7290	. . . . . . 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))))
80		eqid 2738	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
81		eqid 2738	. . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅)
82	6	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑅 ∈ Ring)
83	3	psrbaglefi 21135	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin)
84	77, 83	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ∈ Fin)
85	42	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
86		simplr 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑥 ∈ 𝐷)
87	3	psrbagf 21121	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐷 → 𝑘:𝐼⟶ℕ0)
88	77, 87	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0)
89	76	simprd 496	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑘 ∘r ≤ 𝑥)
90	3	psrbagcon 21133	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑘:𝐼⟶ℕ0 ∧ 𝑘 ∘r ≤ 𝑥) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥))
91	86, 88, 89, 90	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → ((𝑥 ∘f − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘f − 𝑘) ∘r ≤ 𝑥))
92	91	simpld 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑥 ∘f − 𝑘) ∈ 𝐷)
93	85, 92	ffvelrnd 6962	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))
94	82	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑅 ∈ Ring)
95	25	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
96		simpr 485	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘})
97		breq1 5077	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑗 → (ℎ ∘r ≤ 𝑘 ↔ 𝑗 ∘r ≤ 𝑘))
98	97	elrab 3624	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘))
99	96, 98	sylib 217	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘r ≤ 𝑘))
100	99	simpld 495	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∈ 𝐷)
101	95, 100	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑋‘𝑗) ∈ (Base‘𝑅))
102	34	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
103	77	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 ∈ 𝐷)
104	100, 45	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗:𝐼⟶ℕ0)
105	99	simprd 496	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 ∘r ≤ 𝑘)
106	3	psrbagcon 21133	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘r ≤ 𝑘) → ((𝑘 ∘f − 𝑗) ∈ 𝐷 ∧ (𝑘 ∘f − 𝑗) ∘r ≤ 𝑘))
107	103, 104, 105, 106	syl3anc 1370	. . . . . . . . . . 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))
108	107	simpld 495	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑘 ∘f − 𝑗) ∈ 𝐷)
109	102, 108	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅))
110	2, 59	ringcl 19800	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘 ∘f − 𝑗)) ∈ (Base‘𝑅)) → ((𝑋‘𝑗)(.r‘𝑅)(𝑌‘(𝑘 ∘f − 𝑗))) ∈ (Base‘𝑅))
111	94, 101, 109, 110	syl3anc 1370	. . . . . . . 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))
112		eqid 2738	. . . . . . . . 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))))
113		fvex 6787	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
114	113	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → (0g‘𝑅) ∈ V)
115	112, 84, 111, 114	fsuppmptdm 9139	. . . . . . . 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))
116	2, 80, 81, 59, 82, 84, 93, 111, 115	gsummulc1 19845	. . . . . . 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))))
117	93	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → (𝑍‘(𝑥 ∘f − 𝑘)) ∈ (Base‘𝑅))
118	2, 59	ringass 19803	. . . . . . . . . . 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)))))
119	94, 101, 109, 117, 118	syl13anc 1371	. . . . . . . . . 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)))))
120	3	psrbagf 21121	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐷 → 𝑥:𝐼⟶ℕ0)
121	120	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
122	121	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥:𝐼⟶ℕ0)
123	122	ffvelrnda 6961	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈ ℕ0)
124	88	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘:𝐼⟶ℕ0)
125	124	ffvelrnda 6961	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
126	104	ffvelrnda 6961	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈ ℕ0)
127		nn0cn 12243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ)
128		nn0cn 12243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘‘𝑧) ∈ ℕ0 → (𝑘‘𝑧) ∈ ℂ)
129		nn0cn 12243	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ)
130		nnncan2 11258	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥‘𝑧) ∈ ℂ ∧ (𝑘‘𝑧) ∈ ℂ ∧ (𝑗‘𝑧) ∈ ℂ) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
131	127, 128, 129, 130	syl3an 1159	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥‘𝑧) ∈ ℕ0 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑗‘𝑧) ∈ ℕ0) → (((𝑥‘𝑧) − (𝑗‘𝑧)) − ((𝑘‘𝑧) − (𝑗‘𝑧))) = ((𝑥‘𝑧) − (𝑘‘𝑧)))
132	123, 125, 126, 131	syl3anc 1370	. . . . . . . . . . . . . . 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)))
133	132	mpteq2dva 5174	. . . . . . . . . . . . . 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))))
134		psrring.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
135	134	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝐼 ∈ 𝑉)
136	135	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝐼 ∈ 𝑉)
137		ovexd 7310	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V)
138		ovexd 7310	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧) − (𝑗‘𝑧)) ∈ V)
139	122	feqmptd 6837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧)))
140	104	feqmptd 6837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
141	136, 123, 126, 139, 140	offval2 7553	. . . . . . . . . . . . . . 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))))
142	124	feqmptd 6837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) ∧ 𝑗 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ 𝑘}) → 𝑘 = (𝑧 ∈ 𝐼 ↦ (𝑘‘𝑧)))
143	136, 125, 126, 142, 140	offval2 7553	. . . . . . . . . . . . . . 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))))
144	136, 137, 138, 141, 143	offval2 7553	. . . . . . . . . . . . . 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)))))
145	136, 123, 125, 139, 142	offval2 7553	. . . . . . . . . . . . . 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))))
146	133, 144, 145	3eqtr4d 2788	. . . . . . . . . . . . 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))
147	146	fveq2d 6778	. . . . . . . . . . . 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)))
148	147	oveq2d 7291	. . . . . . . . . . 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))))
149	148	oveq2d 7291	. . . . . . . . . 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)))))
150	119, 149	eqtr4d 2781	. . . . . . . . 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))))))
151	150	mpteq2dva 5174	. . . . . . . 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)))))))
152	151	oveq2d 7291	. . . . . . 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))))))))
153	79, 116, 152	3eqtr2d 2784	. . . . . 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))))))))
154	153	mpteq2dva 5174	. . . . 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)))))))))
155	154	oveq2d 7291	. . . 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))))))))))
156	8	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑌 ∈ 𝐵)
157	10	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → 𝑍 ∈ 𝐵)
158	1, 4, 59, 5, 3, 156, 157, 50	psrmulval 21155	. . . . . . . 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))))))
159	158	oveq2d 7291	. . . . . . 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)))))))
160	3	psrbaglefi 21135	. . . . . . . . 9 ⊢ ((𝑥 ∘f − 𝑗) ∈ 𝐷 → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin)
161	50, 160	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘r ≤ 𝑥}) → {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ Fin)
162		ovex 7308	. . . . . . . . . . . . 13 ⊢ (ℕ0 ↑m 𝐼) ∈ V
163	3, 162	rab2ex 5259	. . . . . . . . . . . 12 ⊢ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ∈ V
164	163	mptex 7099	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛)))) ∈ V
165		funmpt 6472	. . . . . . . . . . 11 ⊢ Fun (𝑛 ∈ {ℎ ∈ 𝐷 ∣ ℎ ∘r ≤ (𝑥 ∘f − 𝑗)} ↦ ((𝑌‘𝑛)(.r‘𝑅)(𝑍‘((𝑥 ∘f − 𝑗) ∘f − 𝑛))))
166	164, 165, 113	3pm3.2i 1338	. . . . . . . . . 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)
167	166	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))
168		suppssdm 7993	. . . . . . . . . . 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))))
169		eqid 2738	. . . . . . . . . . . 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))))
170	169	dmmptss 6144	. . . . . . . . . . 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)}
171	168, 170	sstri 3930	. . . . . . . . . 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)}
172	171	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)})
173		suppssfifsupp 9143	. . . . . . . . 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))
174	167, 161, 172, 173	syl12anc 834	. . . . . . . 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))
175	2, 80, 81, 59, 23, 161, 32, 61, 174	gsummulc2 19846	. . . . . . 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)))))))
176	159, 175	eqtr4d 2781	. . . . . 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)))))))
177	176	mpteq2dva 5174	. . . . 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))))))))
178	177	oveq2d 7291	. . . 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)))))))))
179	70, 155, 178	3eqtr4d 2788	. . 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))))))
180	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
181	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ∈ 𝐵)
182	1, 4, 59, 5, 3, 180, 181, 19	psrmulval 21155	. . 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))))))
183	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑋 ∈ 𝐵)
184	14	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
185	1, 4, 59, 5, 3, 183, 184, 19	psrmulval 21155	. . 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))))))
186	179, 182, 185	3eqtr4d 2788	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
187	13, 17, 186	eqfnfvd 6912	1 ⊢ (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   ↦ cmpt 5157  ^◡ccnv 5588  dom cdm 5589   “ cima 5592  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531   ∘_r cofr 7532   supp csupp 7977   ↑_m cmap 8615  Fincfn 8733   finSupp cfsupp 9128  ℂcc 10869   ≤ cle 11010   − cmin 11205  ℕcn 11973  ℕ₀cn0 12233  Basecbs 16912  +_gcplusg 16962  ._rcmulr 16963  0_gc0g 17150   Σ_g cgsu 17151  CMndccmn 19386  Ringcrg 19783   mPwSer cmps 21107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-ofr 7534  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-tset 16981  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-grp 18580  df-minusg 18581  df-mulg 18701  df-ghm 18832  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-psr 21112

This theorem is referenced by:  psrring  21180