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Theorem cpmatel2 22944

Description: Another property of a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.) (Proof shortened by AV, 27-Nov-2019.)

**Hypotheses**
Ref	Expression
cpmat.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmat.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
cpmat.b	⊢ 𝐵 = (Base‘𝐶)
cpmatel2.k	⊢ 𝐾 = (Base‘𝑅)
cpmatel2.a	⊢ 𝐴 = (algSc‘𝑃)

**Assertion**
Ref	Expression
cpmatel2	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘)))

Distinct variable groups: 𝑖,𝑁,𝑗,𝑘 𝑅,𝑖,𝑗,𝑘 𝑖,𝑀,𝑗,𝑘 𝐴,𝑘 𝐵,𝑖,𝑗 𝑘,𝐾 𝑃,𝑘 Allowed substitution hints: 𝐴(𝑖, 𝑗) 𝐵(𝑘) 𝐶(𝑖, 𝑗, 𝑘) 𝑃(𝑖, 𝑗) 𝑆(𝑖, 𝑗, 𝑘) 𝐾(𝑖, 𝑗)

Proof of Theorem cpmatel2 Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmat.s	. . 3 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		cpmat.p	. . 3 ⊢ 𝑃 = (Poly₁‘𝑅)
3		cpmat.c	. . 3 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4		cpmat.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
5	1, 2, 3, 4	cpmatel 22942	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑙) = (0_g‘𝑅)))
6		simpl2 1211	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
7		eqid 2762	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
8		simprl 783	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
9		simprr 785	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
10		simpl3 1212	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
11	3, 7, 4, 8, 9, 10	matecld 22654	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
12		cpmatel2.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
13		eqid 2762	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
14		cpmatel2.a	. . . . . 6 ⊢ 𝐴 = (algSc‘𝑃)
15	12, 13, 2, 7, 14	cply1coe0bi 22533	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑃)) → (∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘) ↔ ∀𝑙 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑙) = (0_g‘𝑅)))
16	6, 11, 15	syl2anc 596	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘) ↔ ∀𝑙 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑙) = (0_g‘𝑅)))
17	16	bicomd 226	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (∀𝑙 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑙) = (0_g‘𝑅) ↔ ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘)))
18	17	2ralbidva 3226	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑙) = (0_g‘𝑅) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘)))
19	5, 18	bitrd 282	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 ∃wrex 3088 ‘cfv 6537 (class class class)co 7417 Fincfn 8956 ℕcn 12261 Basecbs 17307 0_gc0g 17530 Ringcrg 20378 algSccascl 22073 Poly₁cpl1 22408 coe₁cco1 22409 Mat cmat 22635 ConstPolyMat ccpmat 22934

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-ot 4596 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-sup 9416 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-fz 13566 df-fzo 13714 df-seq 14070 df-hash 14399 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-hom 17372 df-cco 17373 df-0g 17532 df-gsum 17533 df-prds 17538 df-pws 17540 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-mulg 19197 df-subg 19252 df-ghm 19347 df-cntz 19450 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-srg 20332 df-ring 20380 df-subrng 20714 df-subrg 20738 df-lmod 21052 df-lss 21122 df-sra 21363 df-rgmod 21364 df-dsmm 21951 df-frlm 21966 df-ascl 22076 df-psr 22130 df-mvr 22131 df-mpl 22132 df-opsr 22134 df-psr1 22411 df-vr1 22412 df-ply1 22413 df-coe1 22414 df-mat 22636 df-cpmat 22937

This theorem is used by: cpmatelimp2 22945 cpmatacl 22947 cpmatinvcl 22948

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