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Theorem psrass1 22001
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 (𝜑𝐼𝑉)
psrring.r (𝜑𝑅 ∈ Ring)
psrass.d 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrass.t × = (.r𝑆)
psrass.b 𝐵 = (Base‘𝑆)
psrass.x (𝜑𝑋𝐵)
psrass.y (𝜑𝑌𝐵)
psrass.z (𝜑𝑍𝐵)
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 𝑗 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrring.s . . . 4 𝑆 = (𝐼 mPwSer 𝑅)
2 eqid 2734 . . . 4 (Base‘𝑅) = (Base‘𝑅)
3 psrass.d . . . 4 𝐷 = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
4 psrass.b . . . 4 𝐵 = (Base‘𝑆)
5 psrass.t . . . . 5 × = (.r𝑆)
6 psrring.r . . . . 5 (𝜑𝑅 ∈ Ring)
7 psrass.x . . . . . 6 (𝜑𝑋𝐵)
8 psrass.y . . . . . 6 (𝜑𝑌𝐵)
91, 4, 5, 6, 7, 8psrmulcl 21983 . . . . 5 (𝜑 → (𝑋 × 𝑌) ∈ 𝐵)
10 psrass.z . . . . 5 (𝜑𝑍𝐵)
111, 4, 5, 6, 9, 10psrmulcl 21983 . . . 4 (𝜑 → ((𝑋 × 𝑌) × 𝑍) ∈ 𝐵)
121, 2, 3, 4, 11psrelbas 21971 . . 3 (𝜑 → ((𝑋 × 𝑌) × 𝑍):𝐷⟶(Base‘𝑅))
1312ffnd 6737 . 2 (𝜑 → ((𝑋 × 𝑌) × 𝑍) Fn 𝐷)
141, 4, 5, 6, 8, 10psrmulcl 21983 . . . . 5 (𝜑 → (𝑌 × 𝑍) ∈ 𝐵)
151, 4, 5, 6, 7, 14psrmulcl 21983 . . . 4 (𝜑 → (𝑋 × (𝑌 × 𝑍)) ∈ 𝐵)
161, 2, 3, 4, 15psrelbas 21971 . . 3 (𝜑 → (𝑋 × (𝑌 × 𝑍)):𝐷⟶(Base‘𝑅))
1716ffnd 6737 . 2 (𝜑 → (𝑋 × (𝑌 × 𝑍)) Fn 𝐷)
18 eqid 2734 . . . . 5 {𝑔𝐷𝑔r𝑥} = {𝑔𝐷𝑔r𝑥}
19 simpr 484 . . . . 5 ((𝜑𝑥𝐷) → 𝑥𝐷)
206ringcmnd 20297 . . . . . 6 (𝜑𝑅 ∈ CMnd)
2120adantr 480 . . . . 5 ((𝜑𝑥𝐷) → 𝑅 ∈ CMnd)
22 eqid 2734 . . . . . . 7 (.r𝑅) = (.r𝑅)
236ad3antrrr 730 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑅 ∈ Ring)
241, 2, 3, 4, 7psrelbas 21971 . . . . . . . . . 10 (𝜑𝑋:𝐷⟶(Base‘𝑅))
2524ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
26 simpr 484 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗 ∈ {𝑔𝐷𝑔r𝑥})
27 breq1 5150 . . . . . . . . . . . 12 (𝑔 = 𝑗 → (𝑔r𝑥𝑗r𝑥))
2827elrab 3694 . . . . . . . . . . 11 (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑗𝐷𝑗r𝑥))
2926, 28sylib 218 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗𝐷𝑗r𝑥))
3029simpld 494 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗𝐷)
3125, 30ffvelcdmd 7104 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑋𝑗) ∈ (Base‘𝑅))
3231adantr 480 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑋𝑗) ∈ (Base‘𝑅))
331, 2, 3, 4, 8psrelbas 21971 . . . . . . . . . 10 (𝜑𝑌:𝐷⟶(Base‘𝑅))
3433ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑌:𝐷⟶(Base‘𝑅))
35 simpr 484 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})
36 breq1 5150 . . . . . . . . . . . 12 ( = 𝑛 → (r ≤ (𝑥f𝑗) ↔ 𝑛r ≤ (𝑥f𝑗)))
3736elrab 3694 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↔ (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
3835, 37sylib 218 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑛𝐷𝑛r ≤ (𝑥f𝑗)))
3938simpld 494 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛𝐷)
4034, 39ffvelcdmd 7104 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑌𝑛) ∈ (Base‘𝑅))
411, 2, 3, 4, 10psrelbas 21971 . . . . . . . . . 10 (𝜑𝑍:𝐷⟶(Base‘𝑅))
4241ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑍:𝐷⟶(Base‘𝑅))
43 simplr 769 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
443psrbagf 21955 . . . . . . . . . . . . . . 15 (𝑗𝐷𝑗:𝐼⟶ℕ0)
4530, 44syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗:𝐼⟶ℕ0)
4629simprd 495 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑗r𝑥)
473psrbagcon 21962 . . . . . . . . . . . . . 14 ((𝑥𝐷𝑗:𝐼⟶ℕ0𝑗r𝑥) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
4843, 45, 46, 47syl3anc 1370 . . . . . . . . . . . . 13 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑗) ∈ 𝐷 ∧ (𝑥f𝑗) ∘r𝑥))
4948simpld 494 . . . . . . . . . . . 12 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑗) ∈ 𝐷)
5049adantr 480 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑥f𝑗) ∈ 𝐷)
513psrbagf 21955 . . . . . . . . . . . 12 (𝑛𝐷𝑛:𝐼⟶ℕ0)
5239, 51syl 17 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛:𝐼⟶ℕ0)
5338simprd 495 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → 𝑛r ≤ (𝑥f𝑗))
543psrbagcon 21962 . . . . . . . . . . 11 (((𝑥f𝑗) ∈ 𝐷𝑛:𝐼⟶ℕ0𝑛r ≤ (𝑥f𝑗)) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
5550, 52, 53, 54syl3anc 1370 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (((𝑥f𝑗) ∘f𝑛) ∈ 𝐷 ∧ ((𝑥f𝑗) ∘f𝑛) ∘r ≤ (𝑥f𝑗)))
5655simpld 494 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑥f𝑗) ∘f𝑛) ∈ 𝐷)
5742, 56ffvelcdmd 7104 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) ∈ (Base‘𝑅))
582, 22, 23, 40, 57ringcld 20276 . . . . . . 7 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) ∈ (Base‘𝑅))
592, 22, 23, 32, 58ringcld 20276 . . . . . 6 ((((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)}) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
6059anasss 466 . . . . 5 (((𝜑𝑥𝐷) ∧ (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ∧ 𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)})) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ (Base‘𝑅))
61 fveq2 6906 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑌𝑛) = (𝑌‘(𝑘f𝑗)))
62 oveq2 7438 . . . . . . . 8 (𝑛 = (𝑘f𝑗) → ((𝑥f𝑗) ∘f𝑛) = ((𝑥f𝑗) ∘f − (𝑘f𝑗)))
6362fveq2d 6910 . . . . . . 7 (𝑛 = (𝑘f𝑗) → (𝑍‘((𝑥f𝑗) ∘f𝑛)) = (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))
6461, 63oveq12d 7448 . . . . . 6 (𝑛 = (𝑘f𝑗) → ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))
6564oveq2d 7446 . . . . 5 (𝑛 = (𝑘f𝑗) → ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
663, 18, 19, 2, 21, 60, 65psrass1lem 21969 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))))
677ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑋𝐵)
688ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
69 simpr 484 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘 ∈ {𝑔𝐷𝑔r𝑥})
70 breq1 5150 . . . . . . . . . . . 12 (𝑔 = 𝑘 → (𝑔r𝑥𝑘r𝑥))
7170elrab 3694 . . . . . . . . . . 11 (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↔ (𝑘𝐷𝑘r𝑥))
7269, 71sylib 218 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑘𝐷𝑘r𝑥))
7372simpld 494 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘𝐷)
741, 4, 22, 5, 3, 67, 68, 73psrmulval 21981 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋 × 𝑌)‘𝑘) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))))
7574oveq1d 7445 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))))(.r𝑅)(𝑍‘(𝑥f𝑘))))
76 eqid 2734 . . . . . . . 8 (0g𝑅) = (0g𝑅)
776ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
783psrbaglefi 21963 . . . . . . . . 9 (𝑘𝐷 → {𝐷r𝑘} ∈ Fin)
7973, 78syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r𝑘} ∈ Fin)
8041ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍:𝐷⟶(Base‘𝑅))
81 simplr 769 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑥𝐷)
823psrbagf 21955 . . . . . . . . . . . 12 (𝑘𝐷𝑘:𝐼⟶ℕ0)
8373, 82syl 17 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘:𝐼⟶ℕ0)
8472simprd 495 . . . . . . . . . . 11 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑘r𝑥)
853psrbagcon 21962 . . . . . . . . . . 11 ((𝑥𝐷𝑘:𝐼⟶ℕ0𝑘r𝑥) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
8681, 83, 84, 85syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑥f𝑘) ∈ 𝐷 ∧ (𝑥f𝑘) ∘r𝑥))
8786simpld 494 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑥f𝑘) ∈ 𝐷)
8880, 87ffvelcdmd 7104 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
896ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑅 ∈ Ring)
9024ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
91 simpr 484 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 ∈ {𝐷r𝑘})
92 breq1 5150 . . . . . . . . . . . . 13 ( = 𝑗 → (r𝑘𝑗r𝑘))
9392elrab 3694 . . . . . . . . . . . 12 (𝑗 ∈ {𝐷r𝑘} ↔ (𝑗𝐷𝑗r𝑘))
9491, 93sylib 218 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑗𝐷𝑗r𝑘))
9594simpld 494 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗𝐷)
9690, 95ffvelcdmd 7104 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑋𝑗) ∈ (Base‘𝑅))
9733ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
9873adantr 480 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘𝐷)
9995, 44syl 17 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗:𝐼⟶ℕ0)
10094simprd 495 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗r𝑘)
1013psrbagcon 21962 . . . . . . . . . . . 12 ((𝑘𝐷𝑗:𝐼⟶ℕ0𝑗r𝑘) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
10298, 99, 100, 101syl3anc 1370 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑘f𝑗) ∈ 𝐷 ∧ (𝑘f𝑗) ∘r𝑘))
103102simpld 494 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) ∈ 𝐷)
10497, 103ffvelcdmd 7104 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅))
1052, 22, 89, 96, 104ringcld 20276 . . . . . . . 8 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))) ∈ (Base‘𝑅))
106 eqid 2734 . . . . . . . . 9 (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗))))
107 fvex 6919 . . . . . . . . . 10 (0g𝑅) ∈ V
108107a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (0g𝑅) ∈ V)
109106, 79, 105, 108fsuppmptdm 9413 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))) finSupp (0g𝑅))
1102, 76, 22, 77, 79, 88, 105, 109gsummulc1 20329 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))))) = ((𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))))(.r𝑅)(𝑍‘(𝑥f𝑘))))
11188adantr 480 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))
1122, 22ringass 20270 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((𝑋𝑗) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑘f𝑗)) ∈ (Base‘𝑅) ∧ (𝑍‘(𝑥f𝑘)) ∈ (Base‘𝑅))) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
11389, 96, 104, 111, 112syl13anc 1371 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
1143psrbagf 21955 . . . . . . . . . . . . . . . . . 18 (𝑥𝐷𝑥:𝐼⟶ℕ0)
115114ad3antlr 731 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥:𝐼⟶ℕ0)
116115ffvelcdmda 7103 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑥𝑧) ∈ ℕ0)
11783adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘:𝐼⟶ℕ0)
118117ffvelcdmda 7103 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
11999ffvelcdmda 7103 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (𝑗𝑧) ∈ ℕ0)
120 nn0cn 12533 . . . . . . . . . . . . . . . . 17 ((𝑥𝑧) ∈ ℕ0 → (𝑥𝑧) ∈ ℂ)
121 nn0cn 12533 . . . . . . . . . . . . . . . . 17 ((𝑘𝑧) ∈ ℕ0 → (𝑘𝑧) ∈ ℂ)
122 nn0cn 12533 . . . . . . . . . . . . . . . . 17 ((𝑗𝑧) ∈ ℕ0 → (𝑗𝑧) ∈ ℂ)
123 nnncan2 11543 . . . . . . . . . . . . . . . . 17 (((𝑥𝑧) ∈ ℂ ∧ (𝑘𝑧) ∈ ℂ ∧ (𝑗𝑧) ∈ ℂ) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
124120, 121, 122, 123syl3an 1159 . . . . . . . . . . . . . . . 16 (((𝑥𝑧) ∈ ℕ0 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑗𝑧) ∈ ℕ0) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
125116, 118, 119, 124syl3anc 1370 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧))) = ((𝑥𝑧) − (𝑘𝑧)))
126125mpteq2dva 5247 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
127 psrring.i . . . . . . . . . . . . . . . 16 (𝜑𝐼𝑉)
128127ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝐼𝑉)
129 ovexd 7465 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑥𝑧) − (𝑗𝑧)) ∈ V)
130 ovexd 7465 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) ∧ 𝑧𝐼) → ((𝑘𝑧) − (𝑗𝑧)) ∈ V)
131115feqmptd 6976 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑥 = (𝑧𝐼 ↦ (𝑥𝑧)))
13299feqmptd 6976 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑗 = (𝑧𝐼 ↦ (𝑗𝑧)))
133128, 116, 119, 131, 132offval2 7716 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑗) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑗𝑧))))
134117feqmptd 6976 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → 𝑘 = (𝑧𝐼 ↦ (𝑘𝑧)))
135128, 118, 119, 134, 132offval2 7716 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑘f𝑗) = (𝑧𝐼 ↦ ((𝑘𝑧) − (𝑗𝑧))))
136128, 129, 130, 133, 135offval2 7716 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑧𝐼 ↦ (((𝑥𝑧) − (𝑗𝑧)) − ((𝑘𝑧) − (𝑗𝑧)))))
137128, 116, 118, 131, 134offval2 7716 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑥f𝑘) = (𝑧𝐼 ↦ ((𝑥𝑧) − (𝑘𝑧))))
138126, 136, 1373eqtr4d 2784 . . . . . . . . . . . . 13 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑥f𝑗) ∘f − (𝑘f𝑗)) = (𝑥f𝑘))
139138fveq2d 6910 . . . . . . . . . . . 12 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))) = (𝑍‘(𝑥f𝑘)))
140139oveq2d 7446 . . . . . . . . . . 11 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))) = ((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘))))
141140oveq2d 7446 . . . . . . . . . 10 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘(𝑥f𝑘)))))
142113, 141eqtr4d 2777 . . . . . . . . 9 ((((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) ∧ 𝑗 ∈ {𝐷r𝑘}) → (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))) = ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))
143142mpteq2dva 5247 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘)))) = (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))))
144143oveq2d 7446 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ (((𝑋𝑗)(.r𝑅)(𝑌‘(𝑘f𝑗)))(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))
14575, 110, 1443eqtr2d 2780 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑘 ∈ {𝑔𝐷𝑔r𝑥}) → (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))) = (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))
146145mpteq2dva 5247 . . . . 5 ((𝜑𝑥𝐷) → (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘)))) = (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗)))))))))
147146oveq2d 7446 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑗 ∈ {𝐷r𝑘} ↦ ((𝑋𝑗)(.r𝑅)((𝑌‘(𝑘f𝑗))(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f − (𝑘f𝑗))))))))))
1488ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑌𝐵)
14910ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑍𝐵)
1501, 4, 22, 5, 3, 148, 149, 49psrmulval 21981 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑌 × 𝑍)‘(𝑥f𝑗)) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))
151150oveq2d 7446 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
1526ad2antrr 726 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → 𝑅 ∈ Ring)
1533psrbaglefi 21963 . . . . . . . . 9 ((𝑥f𝑗) ∈ 𝐷 → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
15449, 153syl 17 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → {𝐷r ≤ (𝑥f𝑗)} ∈ Fin)
155 ovex 7463 . . . . . . . . . . . . 13 (ℕ0m 𝐼) ∈ V
1563, 155rab2ex 5347 . . . . . . . . . . . 12 {𝐷r ≤ (𝑥f𝑗)} ∈ V
157156mptex 7242 . . . . . . . . . . 11 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V
158 funmpt 6605 . . . . . . . . . . 11 Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
159157, 158, 1073pm3.2i 1338 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V)
160159a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V))
161 suppssdm 8200 . . . . . . . . . . 11 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
162 eqid 2734 . . . . . . . . . . . 12 (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) = (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))
163162dmmptss 6262 . . . . . . . . . . 11 dom (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ⊆ {𝐷r ≤ (𝑥f𝑗)}
164161, 163sstri 4004 . . . . . . . . . 10 ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)}
165164a1i 11 . . . . . . . . 9 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)})
166 suppssfifsupp 9417 . . . . . . . . 9 ((((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∈ V ∧ Fun (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) ∧ (0g𝑅) ∈ V) ∧ ({𝐷r ≤ (𝑥f𝑗)} ∈ Fin ∧ ((𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) supp (0g𝑅)) ⊆ {𝐷r ≤ (𝑥f𝑗)})) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
167160, 154, 165, 166syl12anc 837 . . . . . . . 8 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))) finSupp (0g𝑅))
1682, 76, 22, 152, 154, 31, 58, 167gsummulc2 20330 . . . . . . 7 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))) = ((𝑋𝑗)(.r𝑅)(𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
169151, 168eqtr4d 2777 . . . . . 6 (((𝜑𝑥𝐷) ∧ 𝑗 ∈ {𝑔𝐷𝑔r𝑥}) → ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))) = (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))
170169mpteq2dva 5247 . . . . 5 ((𝜑𝑥𝐷) → (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗)))) = (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛))))))))
171170oveq2d 7446 . . . 4 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ (𝑅 Σg (𝑛 ∈ {𝐷r ≤ (𝑥f𝑗)} ↦ ((𝑋𝑗)(.r𝑅)((𝑌𝑛)(.r𝑅)(𝑍‘((𝑥f𝑗) ∘f𝑛)))))))))
17266, 147, 1713eqtr4d 2784 . . 3 ((𝜑𝑥𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
1739adantr 480 . . . 4 ((𝜑𝑥𝐷) → (𝑋 × 𝑌) ∈ 𝐵)
17410adantr 480 . . . 4 ((𝜑𝑥𝐷) → 𝑍𝐵)
1751, 4, 22, 5, 3, 173, 174, 19psrmulval 21981 . . 3 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = (𝑅 Σg (𝑘 ∈ {𝑔𝐷𝑔r𝑥} ↦ (((𝑋 × 𝑌)‘𝑘)(.r𝑅)(𝑍‘(𝑥f𝑘))))))
1767adantr 480 . . . 4 ((𝜑𝑥𝐷) → 𝑋𝐵)
17714adantr 480 . . . 4 ((𝜑𝑥𝐷) → (𝑌 × 𝑍) ∈ 𝐵)
1781, 4, 22, 5, 3, 176, 177, 19psrmulval 21981 . . 3 ((𝜑𝑥𝐷) → ((𝑋 × (𝑌 × 𝑍))‘𝑥) = (𝑅 Σg (𝑗 ∈ {𝑔𝐷𝑔r𝑥} ↦ ((𝑋𝑗)(.r𝑅)((𝑌 × 𝑍)‘(𝑥f𝑗))))))
179172, 175, 1783eqtr4d 2784 . 2 ((𝜑𝑥𝐷) → (((𝑋 × 𝑌) × 𝑍)‘𝑥) = ((𝑋 × (𝑌 × 𝑍))‘𝑥))
18013, 17, 179eqfnfvd 7053 1 (𝜑 → ((𝑋 × 𝑌) × 𝑍) = (𝑋 × (𝑌 × 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  {crab 3432  Vcvv 3477  wss 3962   class class class wbr 5147  cmpt 5230  ccnv 5687  dom cdm 5688  cima 5691  Fun wfun 6556  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  r cofr 7695   supp csupp 8183  m cmap 8864  Fincfn 8983   finSupp cfsupp 9398  cc 11150  cle 11293  cmin 11489  cn 12263  0cn0 12523  Basecbs 17244  .rcmulr 17298  0gc0g 17485   Σg cgsu 17486  CMndccmn 19812  Ringcrg 20250   mPwSer cmps 21941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-mulg 19098  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-ur 20199  df-ring 20252  df-psr 21946
This theorem is referenced by:  psrring  22007
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