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Theorem smadiadetglem1 22394
Description: Lemma 1 for smadiadetg 22396. (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 mpodifsnif 7526 . . . . 5 (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ (𝑖𝑀𝑗))
2 mpodifsnif 7526 . . . . 5 (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗𝑁 ↦ (𝑖𝑀𝑗))
31, 2eqtr4i 2762 . . . 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 4131 . . . . . 6 (𝑁 ∖ {𝐾}) ⊆ 𝑁
5 ssid 4004 . . . . . 6 𝑁𝑁
64, 5pm3.2i 470 . . . . 5 ((𝑁 ∖ {𝐾}) ⊆ 𝑁𝑁𝑁)
7 resmpo 7531 . . . . 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 resmpo 7531 . . . . 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 2797 . . 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 1135 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑀𝐵)
13 simp3 1137 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝑆 ∈ (Base‘𝑅))
14 simp2 1136 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → 𝐾𝑁)
15 smadiadet.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
16 smadiadet.b . . . . . 6 𝐵 = (Base‘𝐴)
17 eqid 2731 . . . . . 6 (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
18 eqid 2731 . . . . . 6 (0g𝑅) = (0g𝑅)
1915, 16, 17, 18marrepval 22285 . . . . 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 836 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 𝑆, (0g𝑅)), (𝑖𝑀𝑗))))
2120reseq1d 5980 . . 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 20140 . . . . . . 7 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
24 eqid 2731 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
25 eqid 2731 . . . . . . . 8 (1r𝑅) = (1r𝑅)
2624, 25ringidcl 20155 . . . . . . 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 22285 . . . . 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 836 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, (1r𝑅), (0g𝑅)), (𝑖𝑀𝑗))))
3130reseq1d 5980 . . 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 2781 . 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, 25minmar1marrep 22373 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r𝑅)))
3533, 12, 34sylancr 586 . . . . 5 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅)(1r𝑅)))
3635eqcomd 2737 . . . 4 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝑀(𝑁 matRRep 𝑅)(1r𝑅)) = ((𝑁 minMatR1 𝑅)‘𝑀))
3736oveqd 7429 . . 3 ((𝑀𝐵𝐾𝑁𝑆 ∈ (Base‘𝑅)) → (𝐾(𝑀(𝑁 matRRep 𝑅)(1r𝑅))𝐾) = (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾))
3837reseq1d 5980 . 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 2771 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 395  w3a 1086   = wceq 1540  wcel 2105  cdif 3945  wss 3948  ifcif 4528  {csn 4628   × cxp 5674  cres 5678  cfv 6543  (class class class)co 7412  cmpo 7414  Basecbs 17149  0gc0g 17390  1rcur 20076  Ringcrg 20128  CRingccrg 20129   Mat cmat 22128   matRRep cmarrep 22279   maDet cmdat 22307   minMatR1 cminmar1 22356
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mgp 20030  df-ur 20077  df-ring 20130  df-cring 20131  df-mat 22129  df-marrep 22281  df-minmar1 22358
This theorem is referenced by:  smadiadetg  22396
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