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Theorem scmateALT 22552

Description: Alternate proof of scmate 22550: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 22551 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 22551	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )})
7	6	eleq2d 2847	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )}))
8		oveq 7398	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
9	8	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) ↔ (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )))
10	9	2ralbidv 3225	. . . . . . 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 3185	. . . . . 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 3650	. . . . 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 7399	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖𝑀𝑗) = (𝐼𝑀𝑗))
14		eqeq1 2765	. . . . . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) ↔ (𝐼𝑀𝑗) = if(𝐼 = 𝑗, 𝑐, 0 )))
17		oveq2 7400	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼𝑀𝑗) = (𝐼𝑀𝐽))
18		eqeq2 2773	. . . . . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼𝑀𝑗) = if(𝐼 = 𝑗, 𝑐, 0 ) ↔ (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))
21	16, 20	rspc2v 3592	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) = if(𝑖 = 𝑗, 𝑐, 0 ) → (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))
22	21	reximdv 3176	. . . . . . . 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 485	. . . . . 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 244	. . . 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 242	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))))
28	27	ex 416	. 2 ⊢ (𝑁 ∈ Fin → (𝑅 ∈ Ring → (𝑀 ∈ 𝑆 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ∃𝑐 ∈ 𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 )))))
29	28	3imp1 1360	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 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 {crab 3413 ifcif 4479 ‘cfv 6517 (class class class)co 7392 Fincfn 8923 Basecbs 17228 0_gc0g 17451 Ringcrg 20262 Mat cmat 22447 ScMat cscmat 22529

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-fzo 13657 df-seq 14012 df-hash 14341 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-hom 17293 df-cco 17294 df-0g 17453 df-gsum 17454 df-prds 17459 df-pws 17461 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-subg 19148 df-ghm 19237 df-cntz 19340 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-subrg 20599 df-lmod 20909 df-lss 20979 df-sra 21220 df-rgmod 21221 df-dsmm 21764 df-frlm 21779 df-mamu 22431 df-mat 22448 df-scmat 22531

This theorem is referenced by: (None)

W3C validator