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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 37979 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ≠ 0 ) |
7 | 6 | 3ad2ant1 1129 | . 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 38134 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇)) |
17 | 16 | adantr 483 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → (𝑈 ∘ 𝑠) = ( I ↾ 𝑇)) |
18 | coeq1 5728 | . . . . . 6 ⊢ (𝑈 = 0 → (𝑈 ∘ 𝑠) = ( 0 ∘ 𝑠)) | |
19 | simp1 1132 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
20 | simp3l 1197 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → 𝑠 ∈ 𝐸) | |
21 | 1, 2, 3, 4, 5 | tendo0mul 37977 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → ( 0 ∘ 𝑠) = 0 ) |
22 | 19, 20, 21 | syl2anc 586 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → ( 0 ∘ 𝑠) = 0 ) |
23 | 18, 22 | sylan9eqr 2878 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → (𝑈 ∘ 𝑠) = 0 ) |
24 | 17, 23 | eqtr3d 2858 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) ∧ 𝑈 = 0 ) → ( I ↾ 𝑇) = 0 ) |
25 | 24 | ex 415 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (ℎ ∈ 𝑇 ∧ ℎ ≠ ( I ↾ 𝐵)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 )) → (𝑈 = 0 → ( I ↾ 𝑇) = 0 )) |
26 | 25 | necon3d 3037 | . 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ifcif 4467 ↦ cmpt 5146 I cid 5459 ◡ccnv 5554 ↾ cres 5557 ∘ ccom 5559 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 Basecbs 16483 occoc 16573 joincjn 17554 meetcmee 17555 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 trLctrl 37309 TEndoctendo 37903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-undef 7939 df-map 8408 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tendo 37906 |
This theorem is referenced by: erngdvlem4 38142 erngdvlem4-rN 38150 |
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