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Theorem scmateALT 22456

Description: Alternate proof of scmate 22454: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 22455 but is longer and requires more distinct variables. (Contributed by AV, 19-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
scmatmat.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatmat.b	⊢ 𝐵 = (Base‘𝐴)
scmatmat.s	⊢ 𝑆 = (𝑁 ScMat 𝑅)
scmate.k	⊢ 𝐾 = (Base‘𝑅)
scmate.0	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
scmateALT	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑅,𝑐 𝐼,𝑐 𝐽,𝑐 𝐾,𝑐 𝑆,𝑐 𝐵,𝑐 Allowed substitution hints: 𝐴(𝑐) 0 (𝑐)

Proof of Theorem scmateALT Dummy variables 𝑚 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatmat.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		scmatmat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
3		scmatmat.s	. . . . . 6 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
4		scmate.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
5		scmate.0	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
6	1, 2, 3, 4, 5	scmatmats 22455	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )})
7	6	eleq2d 2822	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )}))
8		oveq 7364	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
9	8	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )))
10	9	2ralbidv 3200	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )))
11	10	rexbidv 3160	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) ↔ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )))
12	11	elrab 3646	. . . . 5 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )} ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )))
13		oveq1 7365	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖𝑀𝑗) = (𝐼𝑀𝑗))
14		eqeq1 2740	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖 = 𝑗 ↔ 𝐼 = 𝑗))
15	14	ifbid 4503	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝑗, 𝑐, 0 ) = if(𝐼 = 𝑗, 𝑐, 0 ))
16	13, 15	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) ↔ (𝐼𝑀𝑗) = if(𝐼 = 𝑗, 𝑐, 0 )))
17		oveq2 7366	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼𝑀𝑗) = (𝐼𝑀𝐽))
18		eqeq2 2748	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝐼 = 𝑗 ↔ 𝐼 = 𝐽))
19	18	ifbid 4503	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → if(𝐼 = 𝑗, 𝑐, 0 ) = if(𝐼 = 𝐽, 𝑐, 0 ))
20	17, 19	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼𝑀𝑗) = if(𝐼 = 𝑗, 𝑐, 0 ) ↔ (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))
21	16, 20	rspc2v 3587	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) → (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))
22	21	reximdv 3151	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))
23	22	com12 32	. . . . . . 7 ⊢ (∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))
24	23	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))
25	24	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))))
26	12, 25	biimtrid 242	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )} → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))))
27	7, 26	sylbid 240	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))))
28	27	ex 412	. 2 ⊢ (𝑁 ∈ Fin → (𝑅 ∈ Ring → (𝑀 ∈ 𝑆 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))))
29	28	3imp1 1348	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 {crab 3399 ifcif 4479 ‘cfv 6492 (class class class)co 7358 Fincfn 8883 Basecbs 17136 0_gc0g 17359 Ringcrg 20168 Mat cmat 22351 ScMat cscmat 22433

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-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-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-ring 20170 df-subrg 20503 df-lmod 20813 df-lss 20883 df-sra 21125 df-rgmod 21126 df-dsmm 21687 df-frlm 21702 df-mamu 22335 df-mat 22352 df-scmat 22435

This theorem is referenced by: (None)

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