![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkuv-2N | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma2 (p) function, given 𝑉. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemk2.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemk2.l | ⊢ ≤ = (le‘𝐾) |
cdlemk2.j | ⊢ ∨ = (join‘𝐾) |
cdlemk2.m | ⊢ ∧ = (meet‘𝐾) |
cdlemk2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemk2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemk2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemk2.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemk2.s | ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))))) |
cdlemk2.q | ⊢ 𝑄 = (𝑆‘𝐶) |
cdlemk2.v | ⊢ 𝑉 = (𝑑 ∈ 𝑇 ↦ (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝑑)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑑 ∘ ◡𝐶)))))) |
Ref | Expression |
---|---|
cdlemkuv-2N | ⊢ (𝐺 ∈ 𝑇 → (𝑉‘𝐺) = (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐶)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemk2.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemk2.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemk2.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemk2.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemk2.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemk2.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | cdlemk2.r | . 2 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
8 | cdlemk2.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
9 | cdlemk2.v | . 2 ⊢ 𝑉 = (𝑑 ∈ 𝑇 ↦ (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝑑)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑑 ∘ ◡𝐶)))))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | cdlemksv 36919 | 1 ⊢ (𝐺 ∈ 𝑇 → (𝑉‘𝐺) = (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐶)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ↦ cmpt 4952 ◡ccnv 5341 ∘ ccom 5346 ‘cfv 6123 ℩crio 6865 (class class class)co 6905 Basecbs 16222 lecple 16312 joincjn 17297 meetcmee 17298 Atomscatm 35338 LHypclh 36059 LTrncltrn 36176 trLctrl 36233 |
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-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-riota 6866 df-ov 6908 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |