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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk5auN | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemk1.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemk1.l | ⊢ ≤ = (le‘𝐾) |
cdlemk1.j | ⊢ ∨ = (join‘𝐾) |
cdlemk1.m | ⊢ ∧ = (meet‘𝐾) |
cdlemk1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemk1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemk1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemk1.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemk1.s | ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))))) |
cdlemk1.o | ⊢ 𝑂 = (𝑆‘𝐷) |
Ref | Expression |
---|---|
cdlemk5auN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐷) ∧ (𝐷 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (((𝐷‘𝑃) ∨ (𝑅‘𝐷)) ∧ ((𝐷‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷)))) ≤ ((𝑋‘𝑃) ∨ (𝑅‘(𝑋 ∘ ◡𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemk1.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemk1.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemk1.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemk1.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemk1.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemk1.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | cdlemk1.r | . 2 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
8 | cdlemk1.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | cdlemk5a 36440 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ ((𝑅‘𝐺) ≠ (𝑅‘𝐷) ∧ (𝐷 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (((𝐷‘𝑃) ∨ (𝑅‘𝐷)) ∧ ((𝐷‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷)))) ≤ ((𝑋‘𝑃) ∨ (𝑅‘(𝑋 ∘ ◡𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ↦ cmpt 4762 I cid 5052 ◡ccnv 5142 ↾ cres 5145 ∘ ccom 5147 ‘cfv 5926 ℩crio 6650 (class class class)co 6690 Basecbs 15904 lecple 15995 joincjn 16991 meetcmee 16992 Atomscatm 34868 HLchlt 34955 LHypclh 35588 LTrncltrn 35705 trLctrl 35763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-undef 7444 df-map 7901 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 |
This theorem is referenced by: (None) |
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