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Mirrors > Home > MPE Home > Th. List > nfimdetndef | Structured version Visualization version GIF version |
Description: The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.) |
Ref | Expression |
---|---|
nfimdetndef.d | ⊢ 𝐷 = (𝑁 maDet 𝑅) |
Ref | Expression |
---|---|
nfimdetndef | ⊢ (𝑁 ∉ Fin → 𝐷 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfimdetndef.d | . . 3 ⊢ 𝐷 = (𝑁 maDet 𝑅) | |
2 | eqid 2728 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
3 | eqid 2728 | . . 3 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
4 | eqid 2728 | . . 3 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁)) | |
5 | eqid 2728 | . . 3 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
6 | eqid 2728 | . . 3 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
7 | eqid 2728 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
8 | eqid 2728 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mdetfval 22482 | . 2 ⊢ 𝐷 = (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑅 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) |
10 | df-nel 3043 | . . . . . . 7 ⊢ (𝑁 ∉ Fin ↔ ¬ 𝑁 ∈ Fin) | |
11 | 10 | biimpi 215 | . . . . . 6 ⊢ (𝑁 ∉ Fin → ¬ 𝑁 ∈ Fin) |
12 | 11 | intnanrd 489 | . . . . 5 ⊢ (𝑁 ∉ Fin → ¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
13 | matbas0 22304 | . . . . 5 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝑁 ∉ Fin → (Base‘(𝑁 Mat 𝑅)) = ∅) |
15 | 14 | mpteq1d 5238 | . . 3 ⊢ (𝑁 ∉ Fin → (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑅 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))) |
16 | mpt0 6692 | . . 3 ⊢ (𝑚 ∈ ∅ ↦ (𝑅 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = ∅ | |
17 | 15, 16 | eqtrdi 2784 | . 2 ⊢ (𝑁 ∉ Fin → (𝑚 ∈ (Base‘(𝑁 Mat 𝑅)) ↦ (𝑅 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑁)) ↦ ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝑝)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)𝑚𝑥))))))) = ∅) |
18 | 9, 17 | eqtrid 2780 | 1 ⊢ (𝑁 ∉ Fin → 𝐷 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∉ wnel 3042 Vcvv 3470 ∅c0 4319 ↦ cmpt 5226 ∘ ccom 5677 ‘cfv 6543 (class class class)co 7415 Fincfn 8958 Basecbs 17174 .rcmulr 17228 Σg cgsu 17416 SymGrpcsymg 19315 pmSgncpsgn 19438 mulGrpcmgp 20068 ℤRHomczrh 21419 Mat cmat 22301 maDet cmdat 22480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-1cn 11191 ax-addcl 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-nn 12238 df-slot 17145 df-ndx 17157 df-base 17175 df-mat 22302 df-mdet 22481 |
This theorem is referenced by: mdetfval1 22486 |
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