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Mirrors > Home > MPE Home > Th. List > cramerlem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for cramer 20998. (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 |
---|---|
cramerlem3 | ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cramer.a | . . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | cramer.b | . . 3 ⊢ 𝐵 = (Base‘𝐴) | |
3 | cramer.v | . . 3 ⊢ 𝑉 = ((Base‘𝑅) ↑𝑚 𝑁) | |
4 | cramer.x | . . 3 ⊢ · = (𝑅 maVecMul 〈𝑁, 𝑁〉) | |
5 | cramer.d | . . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
6 | 1, 2, 3, 4, 5 | slesolex 20989 | . 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑣 ∈ 𝑉 (𝑋 · 𝑣) = 𝑌) |
7 | cramer.q | . . . . 5 ⊢ / = (/r‘𝑅) | |
8 | 1, 2, 3, 5, 4, 7 | cramerlem2 20995 | . . . 4 ⊢ ((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))))) |
9 | 8 | 3adant1l 1156 | . . 3 ⊢ (((𝑁 ≠ ∅ ∧ 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))))) |
10 | oveq2 6978 | . . . . . . . 8 ⊢ (𝑧 = 𝑣 → (𝑋 · 𝑧) = (𝑋 · 𝑣)) | |
11 | 10 | eqeq1d 2774 | . . . . . . 7 ⊢ (𝑧 = 𝑣 → ((𝑋 · 𝑧) = 𝑌 ↔ (𝑋 · 𝑣) = 𝑌)) |
12 | oveq2 6978 | . . . . . . . 8 ⊢ (𝑣 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑣) = (𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))) | |
13 | 12 | eqeq1d 2774 | . . . . . . 7 ⊢ (𝑣 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → ((𝑋 · 𝑣) = 𝑌 ↔ (𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌)) |
14 | 11, 13 | rexraleqim 3549 | . . . . . 6 ⊢ ((∃𝑣 ∈ 𝑉 (𝑋 · 𝑣) = 𝑌 ∧ ∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))) → (𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌) |
15 | oveq2 6978 | . . . . . . . . . 10 ⊢ (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = (𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))) | |
16 | 15 | adantl 474 | . . . . . . . . 9 ⊢ (((𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌 ∧ 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) → (𝑋 · 𝑍) = (𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))) |
17 | simpl 475 | . . . . . . . . 9 ⊢ (((𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌 ∧ 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) → (𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌) | |
18 | 16, 17 | eqtrd 2808 | . . . . . . . 8 ⊢ (((𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌 ∧ 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) → (𝑋 · 𝑍) = 𝑌) |
19 | 18 | ex 405 | . . . . . . 7 ⊢ ((𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌 → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌)) |
20 | 19 | a1d 25 | . . . . . 6 ⊢ ((𝑋 · (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) = 𝑌 → (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌))) |
21 | 14, 20 | syl 17 | . . . . 5 ⊢ ((∃𝑣 ∈ 𝑉 (𝑋 · 𝑣) = 𝑌 ∧ ∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))) → (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌))) |
22 | 21 | expcom 406 | . . . 4 ⊢ (∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) → (∃𝑣 ∈ 𝑉 (𝑋 · 𝑣) = 𝑌 → (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌)))) |
23 | 22 | com23 86 | . . 3 ⊢ (∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))) → (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (∃𝑣 ∈ 𝑉 (𝑋 · 𝑣) = 𝑌 → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌)))) |
24 | 9, 23 | mpcom 38 | . 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (∃𝑣 ∈ 𝑉 (𝑋 · 𝑣) = 𝑌 → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌))) |
25 | 6, 24 | mpd 15 | 1 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 ∅c0 4172 〈cop 4441 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 ↑𝑚 cmap 8200 Basecbs 16333 CRingccrg 19015 Unitcui 19106 /rcdvr 19149 Mat cmat 20714 maVecMul cmvmul 20847 matRepV cmatrepV 20864 maDet cmdat 20891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-addf 10408 ax-mulf 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-xor 1489 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-ot 4444 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-tpos 7689 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-sup 8695 df-oi 8763 df-card 9156 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-xnn0 11774 df-z 11788 df-dec 11906 df-uz 12053 df-rp 12199 df-fz 12703 df-fzo 12844 df-seq 13179 df-exp 13239 df-hash 13500 df-word 13667 df-lsw 13720 df-concat 13728 df-s1 13753 df-substr 13798 df-pfx 13847 df-splice 13954 df-reverse 13972 df-s2 14066 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-0g 16565 df-gsum 16566 df-prds 16571 df-pws 16573 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-mhm 17797 df-submnd 17798 df-grp 17888 df-minusg 17889 df-sbg 17890 df-mulg 18006 df-subg 18054 df-ghm 18121 df-gim 18164 df-cntz 18212 df-oppg 18239 df-symg 18261 df-pmtr 18325 df-psgn 18374 df-evpm 18375 df-cmn 18662 df-abl 18663 df-mgp 18957 df-ur 18969 df-srg 18973 df-ring 19016 df-cring 19017 df-oppr 19090 df-dvdsr 19108 df-unit 19109 df-invr 19139 df-dvr 19150 df-rnghom 19184 df-drng 19221 df-subrg 19250 df-lmod 19352 df-lss 19420 df-sra 19660 df-rgmod 19661 df-assa 19800 df-cnfld 20242 df-zring 20314 df-zrh 20347 df-dsmm 20572 df-frlm 20587 df-mamu 20691 df-mat 20715 df-mvmul 20848 df-marrep 20865 df-marepv 20866 df-subma 20884 df-mdet 20892 df-madu 20941 df-minmar1 20942 |
This theorem is referenced by: cramer 20998 |
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