![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleml7 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) |
Ref | Expression |
---|---|
cdleml6.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleml6.j | ⊢ ∨ = (join‘𝐾) |
cdleml6.m | ⊢ ∧ = (meet‘𝐾) |
cdleml6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleml6.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdleml6.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdleml6.p | ⊢ 𝑄 = ((oc‘𝐾)‘𝑊) |
cdleml6.z | ⊢ 𝑍 = ((𝑄 ∨ (𝑅‘𝑏)) ∧ ((ℎ‘𝑄) ∨ (𝑅‘(𝑏 ∘ ◡(𝑠‘ℎ))))) |
cdleml6.y | ⊢ 𝑌 = ((𝑄 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) |
cdleml6.x | ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘(𝑠‘ℎ)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑄) = 𝑌)) |
cdleml6.u | ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if((𝑠‘ℎ) = ℎ, 𝑔, 𝑋)) |
cdleml6.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
cdleml6.o | ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
cdleml7 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ((𝑈 ∘ 𝑠)‘ℎ) = (( I ↾ 𝑇)‘ℎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleml6.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdleml6.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | cdleml6.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
4 | cdleml6.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | cdleml6.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
6 | cdleml6.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
7 | cdleml6.p | . . . 4 ⊢ 𝑄 = ((oc‘𝐾)‘𝑊) | |
8 | cdleml6.z | . . . 4 ⊢ 𝑍 = ((𝑄 ∨ (𝑅‘𝑏)) ∧ ((ℎ‘𝑄) ∨ (𝑅‘(𝑏 ∘ ◡(𝑠‘ℎ))))) | |
9 | cdleml6.y | . . . 4 ⊢ 𝑌 = ((𝑄 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) | |
10 | cdleml6.x | . . . 4 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘(𝑠‘ℎ)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑄) = 𝑌)) | |
11 | cdleml6.u | . . . 4 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if((𝑠‘ℎ) = ℎ, 𝑔, 𝑋)) | |
12 | cdleml6.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
13 | cdleml6.o | . . . 4 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | cdleml6 37057 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 ∈ 𝐸 ∧ (𝑈‘(𝑠‘ℎ)) = ℎ)) |
15 | 14 | simprd 491 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈‘(𝑠‘ℎ)) = ℎ) |
16 | simp1 1172 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 14 | simpld 490 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑈 ∈ 𝐸) |
18 | simp3l 1264 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑠 ∈ 𝐸) | |
19 | simp2 1173 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ℎ ∈ 𝑇) | |
20 | 4, 5, 12 | tendocoval 36842 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸) ∧ ℎ ∈ 𝑇) → ((𝑈 ∘ 𝑠)‘ℎ) = (𝑈‘(𝑠‘ℎ))) |
21 | 16, 17, 18, 19, 20 | syl121anc 1500 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ((𝑈 ∘ 𝑠)‘ℎ) = (𝑈‘(𝑠‘ℎ))) |
22 | fvresi 6692 | . . 3 ⊢ (ℎ ∈ 𝑇 → (( I ↾ 𝑇)‘ℎ) = ℎ) | |
23 | 22 | 3ad2ant2 1170 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (( I ↾ 𝑇)‘ℎ) = ℎ) |
24 | 15, 21, 23 | 3eqtr4d 2872 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ((𝑈 ∘ 𝑠)‘ℎ) = (( I ↾ 𝑇)‘ℎ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∀wral 3118 ifcif 4307 ↦ cmpt 4953 I cid 5250 ◡ccnv 5342 ↾ cres 5345 ∘ ccom 5347 ‘cfv 6124 ℩crio 6866 (class class class)co 6906 Basecbs 16223 occoc 16314 joincjn 17298 meetcmee 17299 HLchlt 35426 LHypclh 36060 LTrncltrn 36177 trLctrl 36234 TEndoctendo 36828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-riotaBAD 35029 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-undef 7665 df-map 8125 df-proset 17282 df-poset 17300 df-plt 17312 df-lub 17328 df-glb 17329 df-join 17330 df-meet 17331 df-p0 17393 df-p1 17394 df-lat 17400 df-clat 17462 df-oposet 35252 df-ol 35254 df-oml 35255 df-covers 35342 df-ats 35343 df-atl 35374 df-cvlat 35398 df-hlat 35427 df-llines 35574 df-lplanes 35575 df-lvols 35576 df-lines 35577 df-psubsp 35579 df-pmap 35580 df-padd 35872 df-lhyp 36064 df-laut 36065 df-ldil 36180 df-ltrn 36181 df-trl 36235 df-tendo 36831 |
This theorem is referenced by: cdleml8 37059 |
Copyright terms: Public domain | W3C validator |