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Mirrors > Home > MPE Home > Th. List > mdetralt2 | Structured version Visualization version GIF version |
Description: The determinant function is alternating regarding rows (matrix is given explicitly by its entries). (Contributed by SO, 16-Jul-2018.) |
Ref | Expression |
---|---|
mdetralt2.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
mdetralt2.k | ⊢ 𝐾 = (Base‘𝑅) |
mdetralt2.z | ⊢ 0 = (0g‘𝑅) |
mdetralt2.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
mdetralt2.n | ⊢ (𝜑 → 𝑁 ∈ Fin) |
mdetralt2.x | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) |
mdetralt2.y | ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾) |
mdetralt2.i | ⊢ (𝜑 → 𝐼 ∈ 𝑁) |
mdetralt2.j | ⊢ (𝜑 → 𝐽 ∈ 𝑁) |
mdetralt2.ij | ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
Ref | Expression |
---|---|
mdetralt2 | ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdetralt2.d | . 2 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
2 | eqid 2733 | . 2 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
3 | eqid 2733 | . 2 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
4 | mdetralt2.z | . 2 ⊢ 0 = (0g‘𝑅) | |
5 | mdetralt2.r | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
6 | mdetralt2.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
7 | mdetralt2.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ Fin) | |
8 | mdetralt2.x | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) | |
9 | 8 | 3adant2 1132 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) |
10 | mdetralt2.y | . . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑌 ∈ 𝐾) | |
11 | 9, 10 | ifcld 4572 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐽, 𝑋, 𝑌) ∈ 𝐾) |
12 | 9, 11 | ifcld 4572 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) ∈ 𝐾) |
13 | 2, 6, 3, 7, 5, 12 | matbas2d 21906 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌))) ∈ (Base‘(𝑁 Mat 𝑅))) |
14 | mdetralt2.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑁) | |
15 | mdetralt2.j | . 2 ⊢ (𝜑 → 𝐽 ∈ 𝑁) | |
16 | mdetralt2.ij | . 2 ⊢ (𝜑 → 𝐼 ≠ 𝐽) | |
17 | eqidd 2734 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) | |
18 | iftrue 4532 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋) | |
19 | 18 | ad2antrl 727 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋) |
20 | csbeq1a 3905 | . . . . . . 7 ⊢ (𝑗 = 𝑤 → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋) | |
21 | 20 | ad2antll 728 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋) |
22 | 19, 21 | eqtrd 2773 | . . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐼 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋) |
23 | eqidd 2734 | . . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐼) → 𝑁 = 𝑁) | |
24 | 14 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐼 ∈ 𝑁) |
25 | simpr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝑤 ∈ 𝑁) | |
26 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑤 ∈ 𝑁) | |
27 | nfcsb1v 3916 | . . . . . . . 8 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋 | |
28 | 27 | nfel1 2920 | . . . . . . 7 ⊢ Ⅎ𝑗⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾 |
29 | 26, 28 | nfim 1900 | . . . . . 6 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾) |
30 | eleq1w 2817 | . . . . . . . 8 ⊢ (𝑗 = 𝑤 → (𝑗 ∈ 𝑁 ↔ 𝑤 ∈ 𝑁)) | |
31 | 30 | anbi2d 630 | . . . . . . 7 ⊢ (𝑗 = 𝑤 → ((𝜑 ∧ 𝑗 ∈ 𝑁) ↔ (𝜑 ∧ 𝑤 ∈ 𝑁))) |
32 | 20 | eleq1d 2819 | . . . . . . 7 ⊢ (𝑗 = 𝑤 → (𝑋 ∈ 𝐾 ↔ ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾)) |
33 | 31, 32 | imbi12d 345 | . . . . . 6 ⊢ (𝑗 = 𝑤 → (((𝜑 ∧ 𝑗 ∈ 𝑁) → 𝑋 ∈ 𝐾) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾))) |
34 | 29, 33, 8 | chvarfv 2234 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑗⦌𝑋 ∈ 𝐾) |
35 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑤 ∈ 𝑁) | |
36 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑗𝐼 | |
37 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑖𝑤 | |
38 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑖⦋𝑤 / 𝑗⦌𝑋 | |
39 | 17, 22, 23, 24, 25, 34, 35, 26, 36, 37, 38, 27 | ovmpodxf 7552 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = ⦋𝑤 / 𝑗⦌𝑋) |
40 | iftrue 4532 | . . . . . . . . 9 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐽, 𝑋, 𝑌) = 𝑋) | |
41 | 40 | ifeq2d 4546 | . . . . . . . 8 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = if(𝑖 = 𝐼, 𝑋, 𝑋)) |
42 | ifid 4566 | . . . . . . . 8 ⊢ if(𝑖 = 𝐼, 𝑋, 𝑋) = 𝑋 | |
43 | 41, 42 | eqtrdi 2789 | . . . . . . 7 ⊢ (𝑖 = 𝐽 → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋) |
44 | 43 | ad2antrl 727 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = 𝑋) |
45 | 20 | ad2antll 728 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → 𝑋 = ⦋𝑤 / 𝑗⦌𝑋) |
46 | 44, 45 | eqtrd 2773 | . . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ (𝑖 = 𝐽 ∧ 𝑗 = 𝑤)) → if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)) = ⦋𝑤 / 𝑗⦌𝑋) |
47 | eqidd 2734 | . . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝑁) ∧ 𝑖 = 𝐽) → 𝑁 = 𝑁) | |
48 | 15 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → 𝐽 ∈ 𝑁) |
49 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑗𝐽 | |
50 | 17, 46, 47, 48, 25, 34, 35, 26, 49, 37, 38, 27 | ovmpodxf 7552 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐽(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = ⦋𝑤 / 𝑗⦌𝑋) |
51 | 39, 50 | eqtr4d 2776 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑁) → (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = (𝐽(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤)) |
52 | 51 | ralrimiva 3147 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑁 (𝐼(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤) = (𝐽(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))𝑤)) |
53 | 1, 2, 3, 4, 5, 13, 14, 15, 16, 52 | mdetralt 22091 | 1 ⊢ (𝜑 → (𝐷‘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐼, 𝑋, if(𝑖 = 𝐽, 𝑋, 𝑌)))) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ⦋csb 3891 ifcif 4526 ‘cfv 6539 (class class class)co 7403 ∈ cmpo 7405 Fincfn 8934 Basecbs 17139 0gc0g 17380 CRingccrg 20047 Mat cmat 21888 maDet cmdat 22067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-addf 11184 ax-mulf 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-xor 1511 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-2o 8461 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-sup 9432 df-oi 9500 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-xnn0 12540 df-z 12554 df-dec 12673 df-uz 12818 df-rp 12970 df-fz 13480 df-fzo 13623 df-seq 13962 df-exp 14023 df-hash 14286 df-word 14460 df-lsw 14508 df-concat 14516 df-s1 14541 df-substr 14586 df-pfx 14616 df-splice 14695 df-reverse 14704 df-s2 14794 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-starv 17207 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 df-ple 17212 df-ds 17214 df-unif 17215 df-hom 17216 df-cco 17217 df-0g 17382 df-gsum 17383 df-prds 17388 df-pws 17390 df-mre 17525 df-mrc 17526 df-acs 17528 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-mhm 18666 df-submnd 18667 df-efmnd 18745 df-grp 18817 df-minusg 18818 df-mulg 18944 df-subg 18996 df-ghm 19083 df-gim 19126 df-cntz 19174 df-oppg 19202 df-symg 19227 df-pmtr 19302 df-psgn 19351 df-evpm 19352 df-cmn 19642 df-abl 19643 df-mgp 19979 df-ur 19996 df-ring 20048 df-cring 20049 df-oppr 20138 df-dvdsr 20159 df-unit 20160 df-invr 20190 df-dvr 20203 df-rnghom 20239 df-drng 20305 df-subrg 20348 df-sra 20772 df-rgmod 20773 df-cnfld 20929 df-zring 21002 df-zrh 21036 df-dsmm 21270 df-frlm 21285 df-mat 21889 df-mdet 22068 |
This theorem is referenced by: mdetero 22093 madurid 22127 |
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