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Hypotheses
Ref Expression
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)))

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)
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)))
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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