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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 1134 | . 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 37264 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
6 | 5 | 3adant3 1128 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → 𝐹:𝐵–1-1-onto→𝐵) |
7 | f1orel 6621 | . . . . . . . 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 6049 | . . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
10 | 8, 9 | sylib 220 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ◡◡𝐹 = 𝐹) |
11 | cnveq 5747 | . . . . . 6 ⊢ (◡𝐹 = ( I ↾ 𝐵) → ◡◡𝐹 = ◡( I ↾ 𝐵)) | |
12 | 10, 11 | sylan9req 2880 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ ◡𝐹 = ( I ↾ 𝐵)) → 𝐹 = ◡( I ↾ 𝐵)) |
13 | cnvresid 6436 | . . . . 5 ⊢ ◡( I ↾ 𝐵) = ( I ↾ 𝐵) | |
14 | 12, 13 | syl6eq 2875 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) ∧ ◡𝐹 = ( I ↾ 𝐵)) → 𝐹 = ( I ↾ 𝐵)) |
15 | 14 | ex 415 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → (◡𝐹 = ( I ↾ 𝐵) → 𝐹 = ( I ↾ 𝐵))) |
16 | 15 | necon3d 3040 | . 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 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 I cid 5462 ◡ccnv 5557 ↾ cres 5560 Rel wrel 5563 –1-1-onto→wf1o 6357 ‘cfv 6358 Basecbs 16486 HLchlt 36490 LHypclh 37124 LTrncltrn 37241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-laut 37129 df-ldil 37244 df-ltrn 37245 |
This theorem is referenced by: cdlemh2 37956 cdlemh 37957 cdlemkfid1N 38061 |
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