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Theorem cpmatmcllem 20745
 Description: Lemma for cpmatmcl 20746. (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 2760 . . . 4 (Base‘𝐶) = (Base‘𝐶)
51, 2, 3, 4cpmatelimp 20739 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑥𝑆 → (𝑥 ∈ (Base‘𝐶) ∧ ∀𝑖𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅))))
61, 2, 3, 4cpmatelimp 20739 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
76adantr 472 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑦𝑆 → (𝑦 ∈ (Base‘𝐶) ∧ ∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))))
8 ralcom 3236 . . . . . . . . . . . . . . . 16 (∀𝑙𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ∀𝑗𝑁𝑙𝑁𝑐 ∈ ℕ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
9 r19.26-2 3203 . . . . . . . . . . . . . . . . . . . . . . 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 3236 . . . . . . . . . . . . . . . . . . . . . . 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 1992 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐(((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁))
13 nfra1 3079 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑐𝑐 ∈ ℕ ∀𝑙𝑁 (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅))
1412, 13nfan 1977 . . . . . . . . . . . . . . . . . . . . . . . . 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 827 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑅 ∈ Ring)
16 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (Base‘𝑃) = (Base‘𝑃)
17 simplrl 819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑖𝑁)
18 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑘𝑁)
19 simplrl 819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑥 ∈ (Base‘𝐶))
2019adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑥 ∈ (Base‘𝐶))
213, 16, 4, 17, 18, 20matecld 20454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑖𝑥𝑘) ∈ (Base‘𝑃))
22 simplrr 820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑗𝑁)
23 simplrr 820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → 𝑦 ∈ (Base‘𝐶))
2423adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑦 ∈ (Base‘𝐶))
253, 16, 4, 18, 22, 24matecld 20454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑘𝑦𝑗) ∈ (Base‘𝑃))
2615, 21, 25jca32 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → (𝑅 ∈ Ring ∧ ((𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃))))
2726adantlr 753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6822 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑙 = 𝑘 → (𝑖𝑥𝑙) = (𝑖𝑥𝑘))
2928fveq2d 6357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑖𝑥𝑙)) = (coe1‘(𝑖𝑥𝑘)))
3029fveq1d 6355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑖𝑥𝑙))‘𝑐) = ((coe1‘(𝑖𝑥𝑘))‘𝑐))
3130eqeq1d 2762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅)))
32 oveq1 6821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑙 = 𝑘 → (𝑙𝑦𝑗) = (𝑘𝑦𝑗))
3332fveq2d 6357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (𝑙 = 𝑘 → (coe1‘(𝑙𝑦𝑗)) = (coe1‘(𝑘𝑦𝑗)))
3433fveq1d 6355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (𝑙 = 𝑘 → ((coe1‘(𝑙𝑦𝑗))‘𝑐) = ((coe1‘(𝑘𝑦𝑗))‘𝑐))
3534eqeq1d 2762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑙 = 𝑘 → (((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅) ↔ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅)))
3631, 35anbi12d 749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑙 = 𝑘 → ((((coe1‘(𝑖𝑥𝑙))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑙𝑦𝑗))‘𝑐) = (0g𝑅)) ↔ (((coe1‘(𝑖𝑥𝑘))‘𝑐) = (0g𝑅) ∧ ((coe1‘(𝑘𝑦𝑗))‘𝑐) = (0g𝑅))))
3736rspcva 3447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (0g𝑅) = (0g𝑅)
46 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (.r𝑃) = (.r𝑃)
472, 16, 45, 46cply1mul 19886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3070 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4896 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6830 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
5453anim2i 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Mnd))
5554ancomd 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin))
5645gsumz 17595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑅 ∈ Mnd ∧ 𝑁 ∈ Fin) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5755, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑅 Σg (𝑘𝑁 ↦ (0g𝑅))) = (0g𝑅))
5857ad4antr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . . 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 3095 . . . . . . . . . . . . . . . . . . . . . . . 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 827 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑅 ∈ Ring)
63 nnnn0 11511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 ∈ ℕ → 𝑐 ∈ ℕ0)
6463adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑐 ∈ ℕ0)
652ply1ring 19840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
6665ad4antlr 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → 𝑃 ∈ Ring)
6716, 46ringcl 18781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑃 ∈ Ring ∧ (𝑖𝑥𝑘) ∈ (Base‘𝑃) ∧ (𝑘𝑦𝑗) ∈ (Base‘𝑃)) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6866, 21, 25, 67syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑘𝑁) → ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
6968ralrimiva 3104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
7069adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → ∀𝑘𝑁 ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)) ∈ (Base‘𝑃))
71 simp-4l 825 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ (𝑖𝑁𝑗𝑁)) ∧ 𝑐 ∈ ℕ) → 𝑁 ∈ Fin)
722, 16, 62, 64, 70, 71coe1fzgsumd 19894 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3123 . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . . . . 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 3100 . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 455 . . . . . . . . . . . 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 444 . . . . . . . . . . 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 3100 . . . . . . . . . 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 449 . . . . . . . . 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 644 . . . . . . . 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 446 . . . . . . 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 449 . . . 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 446 . . 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 448 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 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050   ↦ cmpt 4881  ‘cfv 6049  (class class class)co 6814  Fincfn 8123  ℕcn 11232  ℕ0cn0 11504  Basecbs 16079  .rcmulr 16164  0gc0g 16322   Σg cgsu 16323  Mndcmnd 17515  Ringcrg 18767  Poly1cpl1 19769  coe1cco1 19770   Mat cmat 20435   ConstPolyMat ccpmat 20730 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-ofr 7064  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-sup 8515  df-oi 8582  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-fzo 12680  df-seq 13016  df-hash 13332  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-sca 16179  df-vsca 16180  df-ip 16181  df-tset 16182  df-ple 16183  df-ds 16186  df-hom 16188  df-cco 16189  df-0g 16324  df-gsum 16325  df-prds 16330  df-pws 16332  df-mre 16468  df-mrc 16469  df-acs 16471  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mhm 17556  df-submnd 17557  df-grp 17646  df-minusg 17647  df-mulg 17762  df-subg 17812  df-ghm 17879  df-cntz 17970  df-cmn 18415  df-abl 18416  df-mgp 18710  df-ur 18722  df-ring 18769  df-subrg 19000  df-sra 19394  df-rgmod 19395  df-psr 19578  df-mpl 19580  df-opsr 19582  df-psr1 19772  df-ply1 19774  df-coe1 19775  df-dsmm 20298  df-frlm 20313  df-mat 20436  df-cpmat 20733 This theorem is referenced by:  cpmatmcl  20746
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