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Mirrors > Home > MPE Home > Th. List > ma1repvcl | Structured version Visualization version GIF version |
Description: Closure of the column replacement function for identity matrices. (Contributed by AV, 15-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) |
Ref | Expression |
---|---|
ma1repvcl | ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → 𝑅 ∈ Ring) | |
2 | marepvcl.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | marepvcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
4 | ma1repvcl.1 | . . . . . 6 ⊢ 1 = (1r‘𝐴) | |
5 | 2 | fveq2i 6845 | . . . . . 6 ⊢ (1r‘𝐴) = (1r‘(𝑁 Mat 𝑅)) |
6 | 4, 5 | eqtri 2764 | . . . . 5 ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) |
7 | 2, 3, 6 | mat1bas 21796 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1 ∈ 𝐵) |
8 | 7 | anim1i 615 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ( 1 ∈ 𝐵 ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁))) |
9 | 3anass 1095 | . . 3 ⊢ (( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) ↔ ( 1 ∈ 𝐵 ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁))) | |
10 | 8, 9 | sylibr 233 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → ( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) |
11 | marepvcl.v | . . 3 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) | |
12 | 2, 3, 11 | marepvcl 21916 | . 2 ⊢ ((𝑅 ∈ Ring ∧ ( 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) |
13 | 1, 10, 12 | syl2anc 584 | 1 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6496 (class class class)co 7356 ↑m cmap 8764 Fincfn 8882 Basecbs 17082 1rcur 19911 Ringcrg 19962 Mat cmat 21752 matRepV cmatrepV 21904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-sup 9377 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-fzo 13567 df-seq 13906 df-hash 14230 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-hom 17156 df-cco 17157 df-0g 17322 df-gsum 17323 df-prds 17328 df-pws 17330 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-mhm 18600 df-submnd 18601 df-grp 18750 df-minusg 18751 df-sbg 18752 df-mulg 18871 df-subg 18923 df-ghm 19004 df-cntz 19095 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-subrg 20218 df-lmod 20322 df-lss 20391 df-sra 20631 df-rgmod 20632 df-dsmm 21136 df-frlm 21151 df-mamu 21731 df-mat 21753 df-marepv 21906 |
This theorem is referenced by: 1marepvmarrepid 21922 1marepvsma1 21930 cramerimplem1 22030 cramerimplem2 22031 cramerimplem3 22032 cramerimp 22033 |
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