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Theorem smadiadetglem1 20405
Description: Lemma 1 for smadiadetg 20407. (Contributed by AV, 13-Feb-2019.)
Hypotheses
Ref Expression
smadiadet.a 𝐴 = (𝑁 Mat 𝑅)
smadiadet.b 𝐵 = (Base‘𝐴)
smadiadet.r 𝑅 ∈ CRing
smadiadet.d 𝐷 = (𝑁 maDet 𝑅)
smadiadet.h 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)
Assertion
Ref Expression
smadiadetglem1 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))

Proof of Theorem smadiadetglem1
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mpt2difsnif 6713 . . . . 5 (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ (𝑖𝑀𝑗))
2 mpt2difsnif 6713 . . . . 5 (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ (𝑖𝑀𝑗))
31, 2eqtr4i 2646 . . . 4 (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗)))
4 difss 3720 . . . . . 6 (𝑁 ∖ {𝐾}) ⊆ 𝑁
5 ssid 3608 . . . . . 6 𝑁𝑁
64, 5pm3.2i 471 . . . . 5 ((𝑁 ∖ {𝐾}) ⊆ 𝑁𝑁𝑁)
7 resmpt2 6718 . . . . 5 (((𝑁 ∖ {𝐾}) ⊆ 𝑁𝑁𝑁) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
86, 7mp1i 13 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
9 resmpt2 6718 . . . . 5 (((𝑁 ∖ {𝐾}) ⊆ 𝑁𝑁𝑁) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
106, 9mp1i 13 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
113, 8, 103eqtr4a 2681 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
12 simp1 1059 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑀𝐵)
13 simp3 1061 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑆 ∈ (Base‘𝑅))
14 simp2 1060 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
15 smadiadet.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
16 smadiadet.b . . . . . 6 𝐵 = (Base‘𝐴)
17 eqid 2621 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
18 eqid 2621 . . . . . 6 (0g𝑅) = (0g𝑅)
1915, 16, 17, 18marrepval 20296 . . . . 5 (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐾𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
2012, 13, 14, 14, 19syl22anc 1324 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
2120reseq1d 5360 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
22 smadiadet.r . . . . . 6 𝑅 ∈ CRing
23 crngring 18486 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24 eqid 2621 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
25 eqid 2621 . . . . . . . 8 (1r𝑅) = (1r𝑅)
2624, 25ringidcl 18496 . . . . . . 7 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
2723, 26syl 17 . . . . . 6 (𝑅 ∈ CRing → (1r𝑅) ∈ (Base‘𝑅))
2822, 27mp1i 13 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (1r𝑅) ∈ (Base‘𝑅))
2915, 16, 17, 18marrepval 20296 . . . . 5 (((𝑀𝐵 ∧ (1r𝑅) ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐾𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
3012, 28, 14, 14, 29syl22anc 1324 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
3130reseq1d 5360 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
3211, 21, 313eqtr4d 2665 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
3322, 23ax-mp 5 . . . . . 6 𝑅 ∈ Ring
3415, 16, 17, 25minmar1marrep 20384 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r𝑅)))
3533, 12, 34sylancr 694 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r𝑅)))
3635eqcomd 2627 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀(𝑁 matRRep 𝑅)(1r𝑅)) = ((𝑁 minMatR1 𝑅)‘𝑀))
3736oveqd 6627 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) = (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾))
3837reseq1d 5360 . 2 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
3932, 38eqtrd 2655 1 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3556  wss 3559  ifcif 4063  {csn 4153   × cxp 5077  cres 5081  cfv 5852  (class class class)co 6610  cmpt2 6612  Basecbs 15788  0gc0g 16028  1rcur 18429  Ringcrg 18475  CRingccrg 18476   Mat cmat 20141   matRRep cmarrep 20290   maDet cmdat 20318   minMatR1 cminmar1 20367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-ndx 15791  df-slot 15792  df-base 15793  df-sets 15794  df-plusg 15882  df-0g 16030  df-mgm 17170  df-sgrp 17212  df-mnd 17223  df-mgp 18418  df-ur 18430  df-ring 18477  df-cring 18478  df-mat 20142  df-marrep 20292  df-minmar1 20369
This theorem is referenced by:  smadiadetg  20407
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