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Mirrors > Home > MPE Home > Th. List > ma1repveval | Structured version Visualization version GIF version |
Description: An entry of an identity matrix with a replaced column. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.) |
Ref | Expression |
---|---|
marepvcl.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
marepvcl.b | ⊢ 𝐵 = (Base‘𝐴) |
marepvcl.v | ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) |
ma1repvcl.1 | ⊢ 1 = (1r‘𝐴) |
mulmarep1el.0 | ⊢ 0 = (0g‘𝑅) |
mulmarep1el.e | ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) |
Ref | Expression |
---|---|
ma1repveval | ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), if(𝐽 = 𝐼, (1r‘𝑅), 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | marepvcl.a | . . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | marepvcl.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐴) | |
3 | 1, 2 | matrcl 20592 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | 3 | simpld 490 | . . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
5 | ma1repvcl.1 | . . . . . . . . . 10 ⊢ 1 = (1r‘𝐴) | |
6 | 1 | fveq2i 6440 | . . . . . . . . . 10 ⊢ (1r‘𝐴) = (1r‘(𝑁 Mat 𝑅)) |
7 | 5, 6 | eqtri 2849 | . . . . . . . . 9 ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) |
8 | 1, 2, 7 | mat1bas 20630 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ 𝐵) |
9 | 8 | expcom 404 | . . . . . . 7 ⊢ (𝑁 ∈ Fin → (𝑅 ∈ Ring → 1 ∈ 𝐵)) |
10 | 4, 9 | syl 17 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑅 ∈ Ring → 1 ∈ 𝐵)) |
11 | 10 | 3ad2ant1 1167 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → (𝑅 ∈ Ring → 1 ∈ 𝐵)) |
12 | 11 | impcom 398 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 1 ∈ 𝐵) |
13 | simpr2 1254 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝐶 ∈ 𝑉) | |
14 | simpr3 1256 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝐾 ∈ 𝑁) | |
15 | 12, 13, 14 | 3jca 1162 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) |
16 | mulmarep1el.e | . . . . . 6 ⊢ 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) | |
17 | 16 | a1i 11 | . . . . 5 ⊢ ((( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)) |
18 | 17 | oveqd 6927 | . . . 4 ⊢ ((( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = (𝐼(( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)𝐽)) |
19 | eqid 2825 | . . . . 5 ⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅) | |
20 | marepvcl.v | . . . . 5 ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) | |
21 | 1, 2, 19, 20 | marepveval 20749 | . . . 4 ⊢ ((( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼 1 𝐽))) |
22 | 18, 21 | eqtrd 2861 | . . 3 ⊢ ((( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼 1 𝐽))) |
23 | 15, 22 | stoic3 1875 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼 1 𝐽))) |
24 | eqid 2825 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
25 | mulmarep1el.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
26 | 4 | 3ad2ant1 1167 | . . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin) |
27 | 26 | 3ad2ant2 1168 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑁 ∈ Fin) |
28 | simp1 1170 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
29 | simp3l 1262 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁) | |
30 | simp3r 1263 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐽 ∈ 𝑁) | |
31 | 1, 24, 25, 27, 28, 29, 30, 5 | mat1ov 20629 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼 1 𝐽) = if(𝐼 = 𝐽, (1r‘𝑅), 0 )) |
32 | eqcom 2832 | . . . . . 6 ⊢ (𝐼 = 𝐽 ↔ 𝐽 = 𝐼) | |
33 | 32 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼 = 𝐽 ↔ 𝐽 = 𝐼)) |
34 | 33 | ifbid 4330 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → if(𝐼 = 𝐽, (1r‘𝑅), 0 ) = if(𝐽 = 𝐼, (1r‘𝑅), 0 )) |
35 | 31, 34 | eqtrd 2861 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼 1 𝐽) = if(𝐽 = 𝐼, (1r‘𝑅), 0 )) |
36 | 35 | ifeq2d 4327 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼 1 𝐽)) = if(𝐽 = 𝐾, (𝐶‘𝐼), if(𝐽 = 𝐼, (1r‘𝑅), 0 ))) |
37 | 23, 36 | eqtrd 2861 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), if(𝐽 = 𝐼, (1r‘𝑅), 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ifcif 4308 ‘cfv 6127 (class class class)co 6910 ↑𝑚 cmap 8127 Fincfn 8228 Basecbs 16229 0gc0g 16460 1rcur 18862 Ringcrg 18908 Mat cmat 20587 matRepV cmatrepV 20738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-sup 8623 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-hom 16336 df-cco 16337 df-0g 16462 df-gsum 16463 df-prds 16468 df-pws 16470 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-subrg 19141 df-lmod 19228 df-lss 19296 df-sra 19540 df-rgmod 19541 df-dsmm 20446 df-frlm 20461 df-mamu 20564 df-mat 20588 df-marepv 20740 |
This theorem is referenced by: mulmarep1el 20753 1marepvmarrepid 20756 |
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