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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 37995 | 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 1537 ∈ wcel 2114 ↦ cmpt 5146 ◡ccnv 5554 ∘ ccom 5559 ‘cfv 6355 ℩crio 7113 (class class class)co 7156 Basecbs 16483 lecple 16572 joincjn 17554 meetcmee 17555 Atomscatm 36414 LHypclh 37135 LTrncltrn 37252 trLctrl 37309 |
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-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 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-iota 6314 df-fun 6357 df-fv 6363 df-riota 7114 df-ov 7159 |
This theorem is referenced by: (None) |
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