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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncnvnid | Structured version Visualization version GIF version |
Description: If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.) |
Ref | Expression |
---|---|
ltrn1o.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrn1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrn1o.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrncnvnid | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ◡𝐹 ≠ ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1130 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → 𝐹 ≠ ( I ↾ 𝐵)) | |
2 | ltrn1o.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾) | |
3 | ltrn1o.h | . . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | ltrn1o.t | . . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | ltrn1o 37142 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
6 | 5 | 3adant3 1124 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → 𝐹:𝐵–1-1-onto→𝐵) |
7 | f1orel 6612 | . . . . . . . 8 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → Rel 𝐹) | |
8 | 6, 7 | syl 17 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → Rel 𝐹) |
9 | dfrel2 6040 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
10 | 8, 9 | sylib 219 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ◡◡𝐹 = 𝐹) |
11 | cnveq 5738 | . . . . . 6 ⊢ (◡𝐹 = ( I ↾ 𝐵) → ◡◡𝐹 = ◡( I ↾ 𝐵)) | |
12 | 10, 11 | sylan9req 2877 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ ◡𝐹 = ( I ↾ 𝐵)) → 𝐹 = ◡( I ↾ 𝐵)) |
13 | cnvresid 6427 | . . . . 5 ⊢ ◡( I ↾ 𝐵) = ( I ↾ 𝐵) | |
14 | 12, 13 | syl6eq 2872 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ ◡𝐹 = ( I ↾ 𝐵)) → 𝐹 = ( I ↾ 𝐵)) |
15 | 14 | ex 413 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → (◡𝐹 = ( I ↾ 𝐵) → 𝐹 = ( I ↾ 𝐵))) |
16 | 15 | necon3d 3037 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → (𝐹 ≠ ( I ↾ 𝐵) → ◡𝐹 ≠ ( I ↾ 𝐵))) |
17 | 1, 16 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ◡𝐹 ≠ ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 I cid 5453 ◡ccnv 5548 ↾ cres 5551 Rel wrel 5554 –1-1-onto→wf1o 6348 ‘cfv 6349 Basecbs 16473 HLchlt 36368 LHypclh 37002 LTrncltrn 37119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8398 df-laut 37007 df-ldil 37122 df-ltrn 37123 |
This theorem is referenced by: cdlemh2 37834 cdlemh 37835 cdlemkfid1N 37939 |
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