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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemksv | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.) |
Ref | Expression |
---|---|
cdlemk.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemk.l | ⊢ ≤ = (le‘𝐾) |
cdlemk.j | ⊢ ∨ = (join‘𝐾) |
cdlemk.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemk.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemk.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemk.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemk.m | ⊢ ∧ = (meet‘𝐾) |
cdlemk.s | ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))))) |
Ref | Expression |
---|---|
cdlemksv | ⊢ (𝐺 ∈ 𝑇 → (𝑆‘𝐺) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6656 | . . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘𝑓) = (𝑅‘𝐺)) | |
2 | 1 | oveq2d 7158 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝑃 ∨ (𝑅‘𝑓)) = (𝑃 ∨ (𝑅‘𝐺))) |
3 | coeq1 5714 | . . . . . . 7 ⊢ (𝑓 = 𝐺 → (𝑓 ∘ ◡𝐹) = (𝐺 ∘ ◡𝐹)) | |
4 | 3 | fveq2d 6660 | . . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑅‘(𝑓 ∘ ◡𝐹)) = (𝑅‘(𝐺 ∘ ◡𝐹))) |
5 | 4 | oveq2d 7158 | . . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))) = ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))) |
6 | 2, 5 | oveq12d 7160 | . . . 4 ⊢ (𝑓 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))) |
7 | 6 | eqeq2d 2832 | . . 3 ⊢ (𝑓 = 𝐺 → ((𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))) ↔ (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))))) |
8 | 7 | riotabidv 7102 | . 2 ⊢ (𝑓 = 𝐺 → (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))))) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))))) |
9 | cdlemk.s | . 2 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))))) | |
10 | riotaex 7104 | . 2 ⊢ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))))) ∈ V | |
11 | 8, 9, 10 | fvmpt 6754 | 1 ⊢ (𝐺 ∈ 𝑇 → (𝑆‘𝐺) = (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5132 ◡ccnv 5540 ∘ ccom 5545 ‘cfv 6341 ℩crio 7099 (class class class)co 7142 Basecbs 16466 lecple 16555 joincjn 17537 meetcmee 17538 Atomscatm 36431 LHypclh 37152 LTrncltrn 37269 trLctrl 37326 |
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 5189 ax-nul 5196 ax-pr 5316 |
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 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-iota 6300 df-fun 6343 df-fv 6349 df-riota 7100 df-ov 7145 |
This theorem is referenced by: cdlemksel 38013 cdlemksv2 38015 cdlemkuvN 38032 cdlemkuel 38033 cdlemkuv2 38035 cdlemkuv-2N 38051 cdlemkuu 38063 |
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