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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 1082 | . . . . 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 35093 | . . . . 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 2620 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = (𝑂‘𝐹) ↔ (𝑈‘𝐹) = ( I ↾ 𝐵))) |
7 | simpl1 1057 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | simpl2 1058 | . . . . 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 35096 | . . . . . 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 476 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → (𝑈‘𝐹) = (𝑂‘𝐹)) | |
15 | simpl3l 1109 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → 𝐹 ∈ 𝑇) | |
16 | simpl3r 1110 | . . . . 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 35130 | . . . . 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 1336 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑈‘𝐹) = (𝑂‘𝐹)) → 𝑈 = 𝑂) |
19 | 18 | ex 449 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = (𝑂‘𝐹) → 𝑈 = 𝑂)) |
20 | 6, 19 | sylbird 249 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → ((𝑈‘𝐹) = ( I ↾ 𝐵) → 𝑈 = 𝑂)) |
21 | fveq1 6102 | . . . 4 ⊢ (𝑈 = 𝑂 → (𝑈‘𝐹) = (𝑂‘𝐹)) | |
22 | 21 | eqeq1d 2612 | . . 3 ⊢ (𝑈 = 𝑂 → ((𝑈‘𝐹) = ( I ↾ 𝐵) ↔ (𝑂‘𝐹) = ( I ↾ 𝐵))) |
23 | 5, 22 | syl5ibrcom 236 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵))) → (𝑈 = 𝑂 → (𝑈‘𝐹) = ( I ↾ 𝐵))) |
24 | 20, 23 | impbid 201 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ‘cfv 5804 Basecbs 15695 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 TEndoctendo 35058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-undef 7286 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 |
This theorem is referenced by: tendoconid 35135 tendotr 35136 cdleml3N 35284 tendospcanN 35330 |
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