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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 39444 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 ∈ 𝐸 ∧ (𝑈‘(𝑠‘ℎ)) = ℎ)) |
15 | 14 | simprd 496 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈‘(𝑠‘ℎ)) = ℎ) |
16 | simp1 1136 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 14 | simpld 495 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑈 ∈ 𝐸) |
18 | simp3l 1201 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑠 ∈ 𝐸) | |
19 | simp2 1137 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ℎ ∈ 𝑇) | |
20 | 4, 5, 12 | tendocoval 39229 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑠 ∈ 𝐸) ∧ ℎ ∈ 𝑇) → ((𝑈 ∘ 𝑠)‘ℎ) = (𝑈‘(𝑠‘ℎ))) |
21 | 16, 17, 18, 19, 20 | syl121anc 1375 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ((𝑈 ∘ 𝑠)‘ℎ) = (𝑈‘(𝑠‘ℎ))) |
22 | fvresi 7119 | . . 3 ⊢ (ℎ ∈ 𝑇 → (( I ↾ 𝑇)‘ℎ) = ℎ) | |
23 | 22 | 3ad2ant2 1134 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ℎ ∈ 𝑇 ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (( I ↾ 𝑇)‘ℎ) = ℎ) |
24 | 15, 21, 23 | 3eqtr4d 2786 | 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ifcif 4486 ↦ cmpt 5188 I cid 5530 ◡ccnv 5632 ↾ cres 5635 ∘ ccom 5637 ‘cfv 6496 ℩crio 7312 (class class class)co 7357 Basecbs 17083 occoc 17141 joincjn 18200 meetcmee 18201 HLchlt 37812 LHypclh 38447 LTrncltrn 38564 trLctrl 38621 TEndoctendo 39215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-riotaBAD 37415 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-undef 8204 df-map 8767 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tendo 39218 |
This theorem is referenced by: cdleml8 39446 |
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