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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo0mulr | Structured version Visualization version GIF version |
Description: Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.) |
Ref | Expression |
---|---|
tendoid0.b | ⊢ 𝐵 = (Base‘𝐾) |
tendoid0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tendoid0.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tendoid0.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
tendoid0.o | ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
tendo0mulr | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ 𝑂) = 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoid0.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | tendoid0.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendoid0.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | cdlemftr0 37182 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑔 ∈ 𝑇 𝑔 ≠ ( I ↾ 𝐵)) |
5 | 4 | adantr 473 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → ∃𝑔 ∈ 𝑇 𝑔 ≠ ( I ↾ 𝐵)) |
6 | simpll 755 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | simplr 757 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → 𝑈 ∈ 𝐸) | |
8 | tendoid0.e | . . . . . 6 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
9 | tendoid0.o | . . . . . 6 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
10 | 1, 2, 3, 8, 9 | tendo0cl 37404 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
11 | 10 | ad2antrr 714 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → 𝑂 ∈ 𝐸) |
12 | 2, 8 | tendococl 37386 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ 𝑂 ∈ 𝐸) → (𝑈 ∘ 𝑂) ∈ 𝐸) |
13 | 6, 7, 11, 12 | syl3anc 1352 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑈 ∘ 𝑂) ∈ 𝐸) |
14 | 9, 1 | tendo02 37401 | . . . . . . 7 ⊢ (𝑔 ∈ 𝑇 → (𝑂‘𝑔) = ( I ↾ 𝐵)) |
15 | 14 | ad2antrl 716 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑂‘𝑔) = ( I ↾ 𝐵)) |
16 | 15 | fveq2d 6501 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑈‘(𝑂‘𝑔)) = (𝑈‘( I ↾ 𝐵))) |
17 | 1, 2, 8 | tendoid 37387 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
18 | 17 | adantr 473 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
19 | 16, 18 | eqtrd 2809 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑈‘(𝑂‘𝑔)) = ( I ↾ 𝐵)) |
20 | simprl 759 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → 𝑔 ∈ 𝑇) | |
21 | 2, 3, 8 | tendocoval 37380 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑂 ∈ 𝐸) ∧ 𝑔 ∈ 𝑇) → ((𝑈 ∘ 𝑂)‘𝑔) = (𝑈‘(𝑂‘𝑔))) |
22 | 6, 7, 11, 20, 21 | syl121anc 1356 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → ((𝑈 ∘ 𝑂)‘𝑔) = (𝑈‘(𝑂‘𝑔))) |
23 | 19, 22, 15 | 3eqtr4d 2819 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → ((𝑈 ∘ 𝑂)‘𝑔) = (𝑂‘𝑔)) |
24 | simpr 477 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) | |
25 | 1, 2, 3, 8 | tendocan 37438 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑈 ∘ 𝑂) ∈ 𝐸 ∧ 𝑂 ∈ 𝐸 ∧ ((𝑈 ∘ 𝑂)‘𝑔) = (𝑂‘𝑔)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑈 ∘ 𝑂) = 𝑂) |
26 | 6, 13, 11, 23, 24, 25 | syl131anc 1364 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵))) → (𝑈 ∘ 𝑂) = 𝑂) |
27 | 5, 26 | rexlimddv 3231 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ 𝑂) = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 ∃wrex 3084 ↦ cmpt 5005 I cid 5308 ↾ cres 5406 ∘ ccom 5408 ‘cfv 6186 Basecbs 16338 HLchlt 35964 LHypclh 36598 LTrncltrn 36715 TEndoctendo 37366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-riotaBAD 35567 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-iin 4792 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-1st 7500 df-2nd 7501 df-undef 7741 df-map 8207 df-proset 17409 df-poset 17427 df-plt 17439 df-lub 17455 df-glb 17456 df-join 17457 df-meet 17458 df-p0 17520 df-p1 17521 df-lat 17527 df-clat 17589 df-oposet 35790 df-ol 35792 df-oml 35793 df-covers 35880 df-ats 35881 df-atl 35912 df-cvlat 35936 df-hlat 35965 df-llines 36112 df-lplanes 36113 df-lvols 36114 df-lines 36115 df-psubsp 36117 df-pmap 36118 df-padd 36410 df-lhyp 36602 df-laut 36603 df-ldil 36718 df-ltrn 36719 df-trl 36773 df-tendo 37369 |
This theorem is referenced by: dib1dim2 37782 diblss 37784 |
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