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Theorem cpmatmcllem 20517
Description: Lemma for cpmatmcl 20518. (Contributed by AV, 18-Nov-2019.)
Hypotheses
Ref Expression
cpmatsrngpmat.s 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmatsrngpmat.p 𝑃 = (Poly1𝑅)
cpmatsrngpmat.c 𝐶 = (𝑁 Mat 𝑃)
Assertion
Ref Expression
cpmatmcllem (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
Distinct variable groups:   𝐶,𝑖,𝑗   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝐶,𝑐   𝑁,𝑐,𝑥,𝑦,𝑖,𝑗   𝑃,𝑐   𝑅,𝑐,𝑥,𝑦   𝑦,𝑆   𝐶,𝑘   𝑘,𝑁,𝑐,𝑖,𝑗,𝑥,𝑦   𝑃,𝑘   𝑅,𝑘
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑖,𝑗)   𝑆(𝑥,𝑖,𝑗,𝑘,𝑐)

Proof of Theorem cpmatmcllem
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 cpmatsrngpmat.s . . . 4 𝑆 = (𝑁 ConstPolyMat 𝑅)
2 cpmatsrngpmat.p . . . 4 𝑃 = (Poly1𝑅)
3 cpmatsrngpmat.c . . . 4 𝐶 = (𝑁 Mat 𝑃)
4 eqid 2621 . . . 4 (Base‘𝐶) = (Base‘𝐶)
51, 2, 3, 4cpmatelimp 20511 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅))))
61, 2, 3, 4cpmatelimp 20511 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
76adantr 481 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
8 ralcom 3096 . . . . . . . . . . . . . . . 16 (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ∀𝑗𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
9 r19.26-2 3063 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑙𝑁𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
10 ralcom 3096 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑙𝑁𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
119, 10bitr3i 266 . . . . . . . . . . . . . . . . . . . . . 22 ((∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
12 nfv 1842 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁))
13 nfra1 2940 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
1412, 13nfan 1827 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑐((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)))
15 simp-4r 807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
16 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Base‘𝑃) = (Base‘𝑃)
17 simplrl 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑖𝑁)
18 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
19 simplrl 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑥 ∈ (Base‘𝐶))
2019adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑥 ∈ (Base‘𝐶))
213, 16, 4, 17, 18, 20matecld 20226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22 simplrr 801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑗𝑁)
23 simplrr 801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑦 ∈ (Base‘𝐶))
2423adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑦 ∈ (Base‘𝐶))
253, 16, 4, 18, 22, 24matecld 20226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑘𝑦𝑗) ∈ (Base‘𝑃))
2615, 21, 25jca32 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
2726adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
28 oveq2 6655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
2928fveq2d 6193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
3029fveq1d 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
3130eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅)))
32 oveq1 6654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑙 = 𝑘 → (𝑙𝑦𝑗) = (𝑘𝑦𝑗))
3332fveq2d 6193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
3433fveq1d 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
3534eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
3631, 35anbi12d 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑙 = 𝑘 → ((((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
3736rspcva 3305 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑘𝑁 ∧ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
3837a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ((𝑘𝑁 ∧ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
3938exp4b 632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑐 ∈ ℕ → (𝑘𝑁 → (∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))))
4039com23 86 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (𝑘𝑁 → (𝑐 ∈ ℕ → (∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))))
4140imp31 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) ∧ 𝑐 ∈ ℕ) → (∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
4241ralimdva 2961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
4342impancom 456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (𝑘𝑁 → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
4443imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) → ∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
45 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (0g𝑅) = (0g𝑅)
46 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (.r𝑃) = (.r𝑃)
472, 16, 45, 46cply1mul 19658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))) → (∀𝑐 ∈ ℕ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅)))
4827, 44, 47sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) → ∀𝑐 ∈ ℕ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅))
4948r19.21bi 2931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑘𝑁) ∧ 𝑐 ∈ ℕ) → ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅))
5049an32s 846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) ∧ 𝑘𝑁) → ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐) = (0g𝑅))
5150mpteq2dva 4742 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐)) = (𝑘𝑁 ↦ (0g𝑅)))
5251oveq2d 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))))
53 ringmnd 18550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
5453anim2i 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
5554ancomd 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
5645gsumz 17368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5755, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5857ad4antr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5952, 58eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) ∧ 𝑐 ∈ ℕ) → (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅))
6059ex 450 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (𝑐 ∈ ℕ → (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
6114, 60ralrimi 2956 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅))
62 simp-4r 807 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
63 nnnn0 11296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
6463adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
652ply1ring 19612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
6665ad4antlr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑃 ∈ Ring)
6716, 46ringcl 18555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6866, 21, 25, 67syl3anc 1325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6968ralrimiva 2965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
7069adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
71 simp-4l 806 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑁 ∈ Fin)
722, 16, 62, 64, 70, 71coe1fzgsumd 19666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))))
7372eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → (((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅) ↔ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
7473ralbidva 2984 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
7574adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → (∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅) ↔ ∀𝑐 ∈ ℕ (𝑅 Σg (𝑘𝑁 ↦ ((coe1‘((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))‘𝑐))) = (0g𝑅)))
7661, 75mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ ∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
7776ex 450 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (∀𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
7811, 77syl5bi 232 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ((∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
7978expd 452 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
8079expr 643 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (𝑗𝑁 → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
8180com23 86 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (𝑗𝑁 → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
8281imp31 448 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) ∧ 𝑗𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
8382ralimdva 2961 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) → (∀𝑗𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
848, 83syl5bi 232 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) ∧ ∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
8584ex 450 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
8685com23 86 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
8786impancom 456 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (𝑖𝑁 → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
8887imp 445 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ∧ 𝑖𝑁) → (∀𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
8988ralimdva 2961 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))
9089ex 450 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9190expr 643 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦 ∈ (Base‘𝐶) → (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
9291impd 447 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
937, 92syld 47 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦𝑆 → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9493com23 86 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9594ex 450 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥 ∈ (Base‘𝐶) → (∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅)))))
9695impd 447 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅)) → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
975, 96syld 47 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑦𝑆 → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))))
9897imp32 449 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  cmpt 4727  cfv 5886  (class class class)co 6647  Fincfn 7952  cn 11017  0cn0 11289  Basecbs 15851  .rcmulr 15936  0gc0g 16094   Σg cgsu 16095  Mndcmnd 17288  Ringcrg 18541  Poly1cpl1 19541  coe1cco1 19542   Mat cmat 20207   ConstPolyMat ccpmat 20502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-ot 4184  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894  df-ofr 6895  df-om 7063  df-1st 7165  df-2nd 7166  df-supp 7293  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-oadd 7561  df-er 7739  df-map 7856  df-pm 7857  df-ixp 7906  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fsupp 8273  df-sup 8345  df-oi 8412  df-card 8762  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-9 11083  df-n0 11290  df-z 11375  df-dec 11491  df-uz 11685  df-fz 12324  df-fzo 12462  df-seq 12797  df-hash 13113  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-mulr 15949  df-sca 15951  df-vsca 15952  df-ip 15953  df-tset 15954  df-ple 15955  df-ds 15958  df-hom 15960  df-cco 15961  df-0g 16096  df-gsum 16097  df-prds 16102  df-pws 16104  df-mre 16240  df-mrc 16241  df-acs 16243  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-mhm 17329  df-submnd 17330  df-grp 17419  df-minusg 17420  df-mulg 17535  df-subg 17585  df-ghm 17652  df-cntz 17744  df-cmn 18189  df-abl 18190  df-mgp 18484  df-ur 18496  df-ring 18543  df-subrg 18772  df-sra 19166  df-rgmod 19167  df-psr 19350  df-mpl 19352  df-opsr 19354  df-psr1 19544  df-ply1 19546  df-coe1 19547  df-dsmm 20070  df-frlm 20085  df-mat 20208  df-cpmat 20505
This theorem is referenced by:  cpmatmcl  20518
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