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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 2736 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
3 | eqid 2736 | . . 3 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
4 | eqid 2736 | . . 3 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁)) | |
5 | eqid 2736 | . . 3 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
6 | eqid 2736 | . . 3 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
7 | eqid 2736 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
8 | eqid 2736 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mdetfval 21935 | . 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 3050 | . . . . . . 7 ⊢ (𝑁 ∉ Fin ↔ ¬ 𝑁 ∈ Fin) | |
11 | 10 | biimpi 215 | . . . . . 6 ⊢ (𝑁 ∉ Fin → ¬ 𝑁 ∈ Fin) |
12 | 11 | intnanrd 490 | . . . . 5 ⊢ (𝑁 ∉ Fin → ¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
13 | matbas0 21757 | . . . . 5 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝑁 ∉ Fin → (Base‘(𝑁 Mat 𝑅)) = ∅) |
15 | 14 | mpteq1d 5200 | . . 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 6643 | . . 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 2792 | . 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 2788 | 1 ⊢ (𝑁 ∉ Fin → 𝐷 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∉ wnel 3049 Vcvv 3445 ∅c0 4282 ↦ cmpt 5188 ∘ ccom 5637 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 Basecbs 17083 .rcmulr 17134 Σg cgsu 17322 SymGrpcsymg 19148 pmSgncpsgn 19271 mulGrpcmgp 19896 ℤRHomczrh 20900 Mat cmat 21754 maDet cmdat 21933 |
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 7672 ax-cnex 11107 ax-1cn 11109 ax-addcl 11111 |
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-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-op 4593 df-uni 4866 df-iun 4956 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-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-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-nn 12154 df-slot 17054 df-ndx 17066 df-base 17084 df-mat 21755 df-mdet 21934 |
This theorem is referenced by: mdetfval1 21939 |
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