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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 22328 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | 3 | simpld 493 | . . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
5 | ma1repvcl.1 | . . . . . . . . . 10 ⊢ 1 = (1r‘𝐴) | |
6 | 1 | fveq2i 6893 | . . . . . . . . . 10 ⊢ (1r‘𝐴) = (1r‘(𝑁 Mat 𝑅)) |
7 | 5, 6 | eqtri 2753 | . . . . . . . . 9 ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) |
8 | 1, 2, 7 | mat1bas 22367 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ 𝐵) |
9 | 8 | expcom 412 | . . . . . . 7 ⊢ (𝑁 ∈ Fin → (𝑅 ∈ Ring → 1 ∈ 𝐵)) |
10 | 4, 9 | syl 17 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑅 ∈ Ring → 1 ∈ 𝐵)) |
11 | 10 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → (𝑅 ∈ Ring → 1 ∈ 𝐵)) |
12 | 11 | impcom 406 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 1 ∈ 𝐵) |
13 | simpr2 1192 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝐶 ∈ 𝑉) | |
14 | simpr3 1193 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝐾 ∈ 𝑁) | |
15 | 12, 13, 14 | 3jca 1125 | . . 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 7431 | . . . 4 ⊢ ((( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = (𝐼(( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)𝐽)) |
19 | eqid 2725 | . . . . 5 ⊢ (𝑁 matRepV 𝑅) = (𝑁 matRepV 𝑅) | |
20 | marepvcl.v | . . . . 5 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) | |
21 | 1, 2, 19, 20 | marepveval 22486 | . . . 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 2765 | . . 3 ⊢ ((( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼 1 𝐽))) |
23 | 15, 22 | stoic3 1770 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶‘𝐼), (𝐼 1 𝐽))) |
24 | eqid 2725 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
25 | mulmarep1el.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
26 | 4 | 3ad2ant1 1130 | . . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝑁 ∈ Fin) |
27 | 26 | 3ad2ant2 1131 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑁 ∈ Fin) |
28 | simp1 1133 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
29 | simp3l 1198 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁) | |
30 | simp3r 1199 | . . . . 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 22366 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼 1 𝐽) = if(𝐼 = 𝐽, (1r‘𝑅), 0 )) |
32 | eqcom 2732 | . . . . . 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 4545 | . . . 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 2765 | . . 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 4542 | . 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 2765 | 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ifcif 4522 ‘cfv 6541 (class class class)co 7414 ↑m cmap 8841 Fincfn 8960 Basecbs 17177 0gc0g 17418 1rcur 20123 Ringcrg 20175 Mat cmat 22323 matRepV cmatrepV 22475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-ot 4631 df-uni 4902 df-int 4943 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-hom 17254 df-cco 17255 df-0g 17420 df-gsum 17421 df-prds 17426 df-pws 17428 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-subrg 20510 df-lmod 20747 df-lss 20818 df-sra 21060 df-rgmod 21061 df-dsmm 21668 df-frlm 21683 df-mamu 22307 df-mat 22324 df-marepv 22477 |
This theorem is referenced by: mulmarep1el 22490 1marepvmarrepid 22493 |
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