cpmatel2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cpmatel2		Structured version Visualization version GIF version

Theorem cpmatel2 22657

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 22655	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑙 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑙) = (0_g‘𝑅)))
6		simpl2 1193	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
7		eqid 2736	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
8		simprl 770	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
9		simprr 772	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑗 ∈ 𝑁)
10		simpl3 1194	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑀 ∈ 𝐵)
11	3, 7, 4, 8, 9, 10	matecld 22370	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖𝑀𝑗) ∈ (Base‘𝑃))
12		cpmatel2.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
13		eqid 2736	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
14		cpmatel2.a	. . . . . 6 ⊢ 𝐴 = (algSc‘𝑃)
15	12, 13, 2, 7, 14	cply1coe0bi 22246	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑃)) → (∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘) ↔ ∀𝑙 ∈ ℕ ((coe₁‘(𝑖𝑀𝑗))‘𝑙) = (0_g‘𝑅)))
16	6, 11, 15	syl2anc 584	. . . 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 223	. . 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 3198	. 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 279	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 ℕcn 12145 Basecbs 17136 0_gc0g 17359 Ringcrg 20168 algSccascl 21807 Poly₁cpl1 22117 coe₁cco1 22118 Mat cmat 22351 ConstPolyMat ccpmat 22647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-ot 4589 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-ghm 19142 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-srg 20122 df-ring 20170 df-subrng 20479 df-subrg 20503 df-lmod 20813 df-lss 20883 df-sra 21125 df-rgmod 21126 df-dsmm 21687 df-frlm 21702 df-ascl 21810 df-psr 21865 df-mvr 21866 df-mpl 21867 df-opsr 21869 df-psr1 22120 df-vr1 22121 df-ply1 22122 df-coe1 22123 df-mat 22352 df-cpmat 22650

This theorem is referenced by: cpmatelimp2 22658 cpmatacl 22660 cpmatinvcl 22661

W3C validator