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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 36132 | 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 1483 ∈ wcel 1990 ↦ cmpt 4729 ◡ccnv 5113 ∘ ccom 5118 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 Basecbs 15857 lecple 15948 joincjn 16944 meetcmee 16945 Atomscatm 34550 LHypclh 35270 LTrncltrn 35387 trLctrl 35445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 |
This theorem is referenced by: (None) |
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