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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 2823 | . . 3 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅) | |
3 | eqid 2823 | . . 3 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅)) | |
4 | eqid 2823 | . . 3 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁)) | |
5 | eqid 2823 | . . 3 ⊢ (ℤRHom‘𝑅) = (ℤRHom‘𝑅) | |
6 | eqid 2823 | . . 3 ⊢ (pmSgn‘𝑁) = (pmSgn‘𝑁) | |
7 | eqid 2823 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
8 | eqid 2823 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mdetfval 21197 | . 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 3126 | . . . . . . 7 ⊢ (𝑁 ∉ Fin ↔ ¬ 𝑁 ∈ Fin) | |
11 | 10 | biimpi 218 | . . . . . 6 ⊢ (𝑁 ∉ Fin → ¬ 𝑁 ∈ Fin) |
12 | 11 | intnanrd 492 | . . . . 5 ⊢ (𝑁 ∉ Fin → ¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
13 | matbas0 21021 | . . . . 5 ⊢ (¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝑁 ∉ Fin → (Base‘(𝑁 Mat 𝑅)) = ∅) |
15 | 14 | mpteq1d 5157 | . . 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 6492 | . . 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 | syl6eq 2874 | . 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 | syl5eq 2870 | 1 ⊢ (𝑁 ∉ Fin → 𝐷 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3125 Vcvv 3496 ∅c0 4293 ↦ cmpt 5148 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 Basecbs 16485 .rcmulr 16568 Σg cgsu 16716 SymGrpcsymg 18497 pmSgncpsgn 18619 mulGrpcmgp 19241 ℤRHomczrh 20649 Mat cmat 21018 maDet cmdat 21195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-slot 16489 df-base 16491 df-mat 21019 df-mdet 21196 |
This theorem is referenced by: mdetfval1 21201 |
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