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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoid0 | Structured version Visualization version GIF version |
Description: A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (Contributed by NM, 1-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 |
---|---|
tendoid0 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = ( I ↾ 𝐵) ↔ 𝑈 = 𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3l 1197 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → 𝐹 ∈ 𝑇) | |
2 | tendoid0.o | . . . . . 6 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
3 | tendoid0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | 2, 3 | tendo02 37925 | . . . . 5 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → (𝑂‘𝐹) = ( I ↾ 𝐵)) |
6 | 5 | eqeq2d 2834 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = (𝑂‘𝐹) ↔ (𝑈‘𝐹) = ( I ↾ 𝐵))) |
7 | simpl1 1187 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | simpl2 1188 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → 𝑈 ∈ 𝐸) | |
9 | tendoid0.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
10 | tendoid0.t | . . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
11 | tendoid0.e | . . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
12 | 3, 9, 10, 11, 2 | tendo0cl 37928 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
13 | 7, 12 | syl 17 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → 𝑂 ∈ 𝐸) |
14 | simpr 487 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → (𝑈‘𝐹) = (𝑂‘𝐹)) | |
15 | simpl3l 1224 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → 𝐹 ∈ 𝑇) | |
16 | simpl3r 1225 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → 𝐹 ≠ ( I ↾ 𝐵)) | |
17 | 3, 9, 10, 11 | tendocan 37962 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑂 ∈ 𝐸 ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → 𝑈 = 𝑂) |
18 | 7, 8, 13, 14, 15, 16, 17 | syl132anc 1384 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → 𝑈 = 𝑂) |
19 | 18 | ex 415 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = (𝑂‘𝐹) → 𝑈 = 𝑂)) |
20 | 6, 19 | sylbird 262 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = ( I ↾ 𝐵) → 𝑈 = 𝑂)) |
21 | fveq1 6671 | . . . 4 ⊢ (𝑈 = 𝑂 → (𝑈‘𝐹) = (𝑂‘𝐹)) | |
22 | 21 | eqeq1d 2825 | . . 3 ⊢ (𝑈 = 𝑂 → ((𝑈‘𝐹) = ( I ↾ 𝐵) ↔ (𝑂‘𝐹) = ( I ↾ 𝐵))) |
23 | 5, 22 | syl5ibrcom 249 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → (𝑈 = 𝑂 → (𝑈‘𝐹) = ( I ↾ 𝐵))) |
24 | 20, 23 | impbid 214 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = ( I ↾ 𝐵) ↔ 𝑈 = 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 ‘cfv 6357 Basecbs 16485 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 TEndoctendo 37890 |
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 ax-un 7463 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 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-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 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-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-iin 4924 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-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-undef 7941 df-map 8410 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tendo 37893 |
This theorem is referenced by: tendoconid 37967 tendotr 37968 cdleml3N 38116 tendospcanN 38161 |
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