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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendo1ne0 | Structured version Visualization version GIF version |
Description: The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.) |
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 |
---|---|
tendo1ne0 | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoid0.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | tendoid0.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | tendoid0.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | cdlemftr0 37719 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑔 ∈ 𝑇 𝑔 ≠ ( I ↾ 𝐵)) |
5 | simp3 1134 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) → 𝑔 ≠ ( I ↾ 𝐵)) | |
6 | fveq1 6669 | . . . . . . . 8 ⊢ (( I ↾ 𝑇) = 𝑂 → (( I ↾ 𝑇)‘𝑔) = (𝑂‘𝑔)) | |
7 | 6 | adantl 484 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ ( I ↾ 𝑇) = 𝑂) → (( I ↾ 𝑇)‘𝑔) = (𝑂‘𝑔)) |
8 | simpl2 1188 | . . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ ( I ↾ 𝑇) = 𝑂) → 𝑔 ∈ 𝑇) | |
9 | fvresi 6935 | . . . . . . . 8 ⊢ (𝑔 ∈ 𝑇 → (( I ↾ 𝑇)‘𝑔) = 𝑔) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ ( I ↾ 𝑇) = 𝑂) → (( I ↾ 𝑇)‘𝑔) = 𝑔) |
11 | tendoid0.o | . . . . . . . . 9 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
12 | 11, 1 | tendo02 37938 | . . . . . . . 8 ⊢ (𝑔 ∈ 𝑇 → (𝑂‘𝑔) = ( I ↾ 𝐵)) |
13 | 8, 12 | syl 17 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ ( I ↾ 𝑇) = 𝑂) → (𝑂‘𝑔) = ( I ↾ 𝐵)) |
14 | 7, 10, 13 | 3eqtr3d 2864 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) ∧ ( I ↾ 𝑇) = 𝑂) → 𝑔 = ( I ↾ 𝐵)) |
15 | 14 | ex 415 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) → (( I ↾ 𝑇) = 𝑂 → 𝑔 = ( I ↾ 𝐵))) |
16 | 15 | necon3d 3037 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) → (𝑔 ≠ ( I ↾ 𝐵) → ( I ↾ 𝑇) ≠ 𝑂)) |
17 | 5, 16 | mpd 15 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 ≠ ( I ↾ 𝐵)) → ( I ↾ 𝑇) ≠ 𝑂) |
18 | 17 | rexlimdv3a 3286 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (∃𝑔 ∈ 𝑇 𝑔 ≠ ( I ↾ 𝐵) → ( I ↾ 𝑇) ≠ 𝑂)) |
19 | 4, 18 | mpd 15 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ↦ cmpt 5146 I cid 5459 ↾ cres 5557 ‘cfv 6355 Basecbs 16483 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 TEndoctendo 37903 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-undef 7939 df-map 8408 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 |
This theorem is referenced by: cdleml9 38135 erngdvlem4 38142 erng1r 38146 erngdvlem4-rN 38150 dvalveclem 38176 dvheveccl 38263 dihord6apre 38407 dihatlat 38485 |
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