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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg2jlemOLDN | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. f preserves join: f(r ∨ s) = f(r) ∨ s, p. 115 10th line from bottom. TODO: Combine with cdlemg2jOLDN 39375? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdlemg2.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemg2.l | ⊢ ≤ = (le‘𝐾) |
cdlemg2.j | ⊢ ∨ = (join‘𝐾) |
cdlemg2.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemg2ex.u | ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊) |
cdlemg2ex.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) |
cdlemg2ex.e | ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdlemg2ex.g | ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) |
Ref | Expression |
---|---|
cdlemg2jlemOLDN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemg2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg2.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemg2.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemg2.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemg2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemg2.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | cdlemg2ex.u | . . 3 ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊) | |
9 | cdlemg2ex.d | . . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) | |
10 | cdlemg2ex.e | . . 3 ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
11 | cdlemg2ex.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) | |
12 | fveq1 6880 | . . . 4 ⊢ (𝐹 = 𝐺 → (𝐹‘(𝑃 ∨ 𝑄)) = (𝐺‘(𝑃 ∨ 𝑄))) | |
13 | fveq1 6880 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃)) | |
14 | fveq1 6880 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑄) = (𝐺‘𝑄)) | |
15 | 13, 14 | oveq12d 7414 | . . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) |
16 | 12, 15 | eqeq12d 2749 | . . 3 ⊢ (𝐹 = 𝐺 → ((𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) ↔ (𝐺‘(𝑃 ∨ 𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄)))) |
17 | vex 3479 | . . . . 5 ⊢ 𝑠 ∈ V | |
18 | eqid 2733 | . . . . . 6 ⊢ ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) | |
19 | 9, 18 | cdleme31sc 39161 | . . . . 5 ⊢ (𝑠 ∈ V → ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊)))) |
20 | 17, 19 | ax-mp 5 | . . . 4 ⊢ ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) |
21 | eqid 2733 | . . . 4 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)) | |
22 | eqid 2733 | . . . 4 ⊢ if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) = if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) | |
23 | eqid 2733 | . . . 4 ⊢ (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))) = (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))) | |
24 | 1, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11 | cdleme42mgN 39265 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐺‘(𝑃 ∨ 𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) |
25 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 24 | cdlemg2ce 39369 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄))) |
26 | 25 | 3com23 1127 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → (𝐹‘(𝑃 ∨ 𝑄)) = ((𝐹‘𝑃) ∨ (𝐹‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ⦋csb 3891 ifcif 4524 class class class wbr 5144 ↦ cmpt 5227 ‘cfv 6535 ℩crio 7351 (class class class)co 7396 Basecbs 17131 lecple 17191 joincjn 18251 meetcmee 18252 Atomscatm 38039 HLchlt 38126 LHypclh 38761 LTrncltrn 38878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-riotaBAD 37729 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7962 df-2nd 7963 df-undef 8245 df-map 8810 df-proset 18235 df-poset 18253 df-plt 18270 df-lub 18286 df-glb 18287 df-join 18288 df-meet 18289 df-p0 18365 df-p1 18366 df-lat 18372 df-clat 18439 df-oposet 37952 df-ol 37954 df-oml 37955 df-covers 38042 df-ats 38043 df-atl 38074 df-cvlat 38098 df-hlat 38127 df-llines 38275 df-lplanes 38276 df-lvols 38277 df-lines 38278 df-psubsp 38280 df-pmap 38281 df-padd 38573 df-lhyp 38765 df-laut 38766 df-ldil 38881 df-ltrn 38882 df-trl 38936 |
This theorem is referenced by: cdlemg2jOLDN 39375 |
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