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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkuvN | Structured version Visualization version GIF version | ||
| Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma1 (p) function 𝑈. (Contributed by NM, 2-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) |
| cdlemk1.u | ⊢ 𝑈 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))))) |
| Ref | Expression |
|---|---|
| cdlemkuvN | ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡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 | cdlemk1.u | . 2 ⊢ 𝑈 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))))) | |
| 10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | cdlemksv 40846 | 1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 ◡ccnv 5684 ∘ ccom 5689 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 meetcmee 18358 Atomscatm 39264 LHypclh 39986 LTrncltrn 40103 trLctrl 40160 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 |
| This theorem is referenced by: (None) |
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