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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleml9 | 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 |
---|---|
cdleml9 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑈 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleml6.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdleml6.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | cdleml6.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | cdleml6.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
5 | cdleml6.o | . . . 4 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
6 | 1, 2, 3, 4, 5 | tendo1ne0 38124 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ 0 ) |
7 | 6 | 3ad2ant1 1130 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ( I ↾ 𝑇) ≠ 0 ) |
8 | cdleml6.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
9 | cdleml6.m | . . . . . . 7 ⊢ ∧ = (meet‘𝐾) | |
10 | cdleml6.r | . . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
11 | cdleml6.p | . . . . . . 7 ⊢ 𝑄 = ((oc‘𝐾)‘𝑊) | |
12 | cdleml6.z | . . . . . . 7 ⊢ 𝑍 = ((𝑄 ∨ (𝑅‘𝑏)) ∧ ((ℎ‘𝑄) ∨ (𝑅‘(𝑏 ∘ ◡(𝑠‘ℎ))))) | |
13 | cdleml6.y | . . . . . . 7 ⊢ 𝑌 = ((𝑄 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) | |
14 | cdleml6.x | . . . . . . 7 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘(𝑠‘ℎ)) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑄) = 𝑌)) | |
15 | cdleml6.u | . . . . . . 7 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if((𝑠‘ℎ) = ℎ, 𝑔, 𝑋)) | |
16 | 1, 8, 9, 2, 3, 10, 11, 12, 13, 14, 15, 4, 5 | cdleml8 38279 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇)) |
17 | 16 | adantr 484 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇)) |
18 | coeq1 5692 | . . . . . 6 ⊢ (𝑈 = 0 → (𝑈 ∘ 𝑠) = ( 0 ∘ 𝑠)) | |
19 | simp1 1133 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
20 | simp3l 1198 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑠 ∈ 𝐸) | |
21 | 1, 2, 3, 4, 5 | tendo0mul 38122 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ( 0 ∘ 𝑠) = 0 ) |
22 | 19, 20, 21 | syl2anc 587 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ( 0 ∘ 𝑠) = 0 ) |
23 | 18, 22 | sylan9eqr 2855 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → (𝑈 ∘ 𝑠) = 0 ) |
24 | 17, 23 | eqtr3d 2835 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → ( I ↾ 𝑇) = 0 ) |
25 | 24 | ex 416 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 = 0 → ( I ↾ 𝑇) = 0 )) |
26 | 25 | necon3d 3008 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (( I ↾ 𝑇) ≠ 0 → 𝑈 ≠ 0 )) |
27 | 7, 26 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑈 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ifcif 4425 ↦ cmpt 5110 I cid 5424 ◡ccnv 5518 ↾ cres 5521 ∘ ccom 5523 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 Basecbs 16475 occoc 16565 joincjn 17546 meetcmee 17547 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 trLctrl 37454 TEndoctendo 38048 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-undef 7922 df-map 8391 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tendo 38051 |
This theorem is referenced by: erngdvlem4 38287 erngdvlem4-rN 38295 |
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