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Mirrors > Home > MPE Home > Th. List > cramerlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for cramer 21294. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.) |
Ref | Expression |
---|---|
cramer.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
cramer.b | ⊢ 𝐵 = (Base‘𝐴) |
cramer.v | ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) |
cramer.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
cramer.x | ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) |
cramer.q | ⊢ / = (/r‘𝑅) |
Ref | Expression |
---|---|
cramerlem1 | ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑅 ∈ CRing) | |
2 | 1 | anim1i 616 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → (𝑅 ∈ CRing ∧ 𝑎 ∈ 𝑁)) |
3 | simpl2 1188 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) | |
4 | pm3.22 462 | . . . . . . 7 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) | |
5 | 4 | 3adant2 1127 | . . . . . 6 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) |
6 | 5 | 3ad2ant3 1131 | . . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) |
7 | 6 | adantr 483 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) |
8 | cramer.a | . . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
9 | cramer.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐴) | |
10 | cramer.v | . . . . 5 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁) | |
11 | eqid 2821 | . . . . 5 ⊢ (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎) = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎) | |
12 | eqid 2821 | . . . . 5 ⊢ ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎) = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎) | |
13 | cramer.x | . . . . 5 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
14 | cramer.d | . . . . 5 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
15 | cramer.q | . . . . 5 ⊢ / = (/r‘𝑅) | |
16 | 8, 9, 10, 11, 12, 13, 14, 15 | cramerimp 21289 | . . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑎 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋))) |
17 | 2, 3, 7, 16 | syl3anc 1367 | . . 3 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋))) |
18 | 17 | ralrimiva 3182 | . 2 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → ∀𝑎 ∈ 𝑁 (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋))) |
19 | elmapfn 8423 | . . . . . 6 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑m 𝑁) → 𝑍 Fn 𝑁) | |
20 | 19, 10 | eleq2s 2931 | . . . . 5 ⊢ (𝑍 ∈ 𝑉 → 𝑍 Fn 𝑁) |
21 | 20 | 3ad2ant2 1130 | . . . 4 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → 𝑍 Fn 𝑁) |
22 | 21 | 3ad2ant3 1131 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 Fn 𝑁) |
23 | 2fveq3 6670 | . . . 4 ⊢ (𝑎 = 𝑖 → (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) = (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖))) | |
24 | 23 | oveq1d 7165 | . . 3 ⊢ (𝑎 = 𝑖 → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) |
25 | ovexd 7185 | . . 3 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)) ∈ V) | |
26 | ovexd 7185 | . . 3 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑖 ∈ 𝑁) → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)) ∈ V) | |
27 | 22, 24, 25, 26 | fnmptfvd 6806 | . 2 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) ↔ ∀𝑎 ∈ 𝑁 (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)))) |
28 | 18, 27 | mpbird 259 | 1 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 〈cop 4567 ↦ cmpt 5139 Fn wfn 6345 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 Basecbs 16477 1rcur 19245 CRingccrg 19292 Unitcui 19383 /rcdvr 19426 Mat cmat 21010 maVecMul cmvmul 21143 matRepV cmatrepV 21160 maDet cmdat 21187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1501 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-reverse 14115 df-s2 14204 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-0g 16709 df-gsum 16710 df-prds 16715 df-pws 16717 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-efmnd 18028 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-gim 18393 df-cntz 18441 df-oppg 18468 df-symg 18490 df-pmtr 18564 df-psgn 18613 df-evpm 18614 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-srg 19250 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-rnghom 19461 df-drng 19498 df-subrg 19527 df-lmod 19630 df-lss 19698 df-sra 19938 df-rgmod 19939 df-cnfld 20540 df-zring 20612 df-zrh 20645 df-dsmm 20870 df-frlm 20885 df-mamu 20989 df-mat 21011 df-mvmul 21144 df-marrep 21161 df-marepv 21162 df-subma 21180 df-mdet 21188 df-minmar1 21238 |
This theorem is referenced by: cramerlem2 21291 cramer 21294 |
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