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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg47a | Structured version Visualization version GIF version |
Description: TODO: fix comment. TODO: Use this above in place of (𝐹‘𝑃) = 𝑃 antecedents? (Contributed by NM, 5-Jun-2013.) |
Ref | Expression |
---|---|
cdlemg46.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemg46.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg46.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemg47a | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1128 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp2r 1192 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → 𝐺 ∈ 𝑇) | |
3 | cdlemg46.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | cdlemg46.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | cdlemg46.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | 3, 4, 5 | ltrn1o 37142 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺:𝐵–1-1-onto→𝐵) |
7 | 1, 2, 6 | syl2anc 584 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → 𝐺:𝐵–1-1-onto→𝐵) |
8 | f1of 6609 | . . . 4 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺:𝐵⟶𝐵) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → 𝐺:𝐵⟶𝐵) |
10 | fcoi1 6546 | . . 3 ⊢ (𝐺:𝐵⟶𝐵 → (𝐺 ∘ ( I ↾ 𝐵)) = 𝐺) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐺 ∘ ( I ↾ 𝐵)) = 𝐺) |
12 | simp3 1130 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → 𝐹 = ( I ↾ 𝐵)) | |
13 | 12 | coeq2d 5727 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐺 ∘ 𝐹) = (𝐺 ∘ ( I ↾ 𝐵))) |
14 | 12 | coeq1d 5726 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐹 ∘ 𝐺) = (( I ↾ 𝐵) ∘ 𝐺)) |
15 | fcoi2 6547 | . . . 4 ⊢ (𝐺:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺) | |
16 | 9, 15 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (( I ↾ 𝐵) ∘ 𝐺) = 𝐺) |
17 | 14, 16 | eqtrd 2856 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐹 ∘ 𝐺) = 𝐺) |
18 | 11, 13, 17 | 3eqtr4rd 2867 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 I cid 5453 ↾ cres 5551 ∘ ccom 5553 ⟶wf 6345 –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: ltrncom 37756 |
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