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Theorem smadiadetglem1 20679
Description: Lemma 1 for smadiadetg 20681. (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 6918 . . . . 5 (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ (𝑖𝑀𝑗))
2 mpt2difsnif 6918 . . . . 5 (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ (𝑖𝑀𝑗))
31, 2eqtr4i 2785 . . . 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 3880 . . . . . 6 (𝑁 ∖ {𝐾}) ⊆ 𝑁
5 ssid 3765 . . . . . 6 𝑁𝑁
64, 5pm3.2i 470 . . . . 5 ((𝑁 ∖ {𝐾}) ⊆ 𝑁𝑁𝑁)
7 resmpt2 6923 . . . . 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 6923 . . . . 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 2820 . . 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 1131 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑀𝐵)
13 simp3 1133 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑆 ∈ (Base‘𝑅))
14 simp2 1132 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
15 smadiadet.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
16 smadiadet.b . . . . . 6 𝐵 = (Base‘𝐴)
17 eqid 2760 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
18 eqid 2760 . . . . . 6 (0g𝑅) = (0g𝑅)
1915, 16, 17, 18marrepval 20570 . . . . 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 1478 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
2120reseq1d 5550 . . 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 18758 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24 eqid 2760 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
25 eqid 2760 . . . . . . . 8 (1r𝑅) = (1r𝑅)
2624, 25ringidcl 18768 . . . . . . 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 20570 . . . . 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 1478 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
3130reseq1d 5550 . . 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 2804 . 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 20658 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r𝑅)))
3533, 12, 34sylancr 698 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r𝑅)))
3635eqcomd 2766 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀(𝑁 matRRep 𝑅)(1r𝑅)) = ((𝑁 minMatR1 𝑅)‘𝑀))
3736oveqd 6830 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) = (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾))
3837reseq1d 5550 . 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 2794 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 383  w3a 1072   = wceq 1632  wcel 2139  cdif 3712  wss 3715  ifcif 4230  {csn 4321   × cxp 5264  cres 5268  cfv 6049  (class class class)co 6813  cmpt2 6815  Basecbs 16059  0gc0g 16302  1rcur 18701  Ringcrg 18747  CRingccrg 18748   Mat cmat 20415   matRRep cmarrep 20564   maDet cmdat 20592   minMatR1 cminmar1 20641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-plusg 16156  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mgp 18690  df-ur 18702  df-ring 18749  df-cring 18750  df-mat 20416  df-marrep 20566  df-minmar1 20643
This theorem is referenced by:  smadiadetg  20681
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