![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg41 | Structured version Visualization version GIF version |
Description: Convert cdlemg40 36526 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.) |
Ref | Expression |
---|---|
cdlemg35.l | ⊢ ≤ = (le‘𝐾) |
cdlemg35.j | ⊢ ∨ = (join‘𝐾) |
cdlemg35.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg35.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg35.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg35.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemg41 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑃 ∨ ((𝐹 ∘ 𝐺)‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ ((𝐹 ∘ 𝐺)‘𝑄)) ∧ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg35.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | cdlemg35.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | cdlemg35.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | cdlemg35.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemg35.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemg35.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | 1, 2, 3, 4, 5, 6 | cdlemg40 36526 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊)) |
8 | simp1 1130 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | simp3 1132 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) | |
10 | simp2ll 1306 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝑃 ∈ 𝐴) | |
11 | 1, 4, 5, 6 | ltrncoval 35953 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑃) = (𝐹‘(𝐺‘𝑃))) |
12 | 8, 9, 10, 11 | syl3anc 1476 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝐹 ∘ 𝐺)‘𝑃) = (𝐹‘(𝐺‘𝑃))) |
13 | 12 | oveq2d 6811 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑃 ∨ ((𝐹 ∘ 𝐺)‘𝑃)) = (𝑃 ∨ (𝐹‘(𝐺‘𝑃)))) |
14 | 13 | oveq1d 6810 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑃 ∨ ((𝐹 ∘ 𝐺)‘𝑃)) ∧ 𝑊) = ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊)) |
15 | simp2rl 1308 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → 𝑄 ∈ 𝐴) | |
16 | 1, 4, 5, 6 | ltrncoval 35953 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑄 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑄) = (𝐹‘(𝐺‘𝑄))) |
17 | 8, 9, 15, 16 | syl3anc 1476 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝐹 ∘ 𝐺)‘𝑄) = (𝐹‘(𝐺‘𝑄))) |
18 | 17 | oveq2d 6811 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑄 ∨ ((𝐹 ∘ 𝐺)‘𝑄)) = (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) |
19 | 18 | oveq1d 6810 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑄 ∨ ((𝐹 ∘ 𝐺)‘𝑄)) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊)) |
20 | 7, 14, 19 | 3eqtr4d 2815 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑃 ∨ ((𝐹 ∘ 𝐺)‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ ((𝐹 ∘ 𝐺)‘𝑄)) ∧ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 class class class wbr 4787 ∘ ccom 5254 ‘cfv 6030 (class class class)co 6795 lecple 16155 joincjn 17151 meetcmee 17152 Atomscatm 35071 HLchlt 35158 LHypclh 35792 LTrncltrn 35909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7099 ax-riotaBAD 34760 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-iin 4658 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6756 df-ov 6798 df-oprab 6799 df-mpt2 6800 df-1st 7318 df-2nd 7319 df-undef 7554 df-map 8014 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-oposet 34984 df-ol 34986 df-oml 34987 df-covers 35074 df-ats 35075 df-atl 35106 df-cvlat 35130 df-hlat 35159 df-llines 35306 df-lplanes 35307 df-lvols 35308 df-lines 35309 df-psubsp 35311 df-pmap 35312 df-padd 35604 df-lhyp 35796 df-laut 35797 df-ldil 35912 df-ltrn 35913 df-trl 35968 |
This theorem is referenced by: ltrnco 36528 |
Copyright terms: Public domain | W3C validator |