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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg2klem | Structured version Visualization version GIF version |
Description: cdleme42keg 40445 with simpler hypotheses. TODO: FIX COMMENT. (Contributed by NM, 22-Apr-2013.) |
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)) |
cdlemg2klem.v | ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
Ref | Expression |
---|---|
cdlemg2klem | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉)) |
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 6921 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃)) | |
13 | fveq1 6921 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑄) = (𝐺‘𝑄)) | |
14 | 12, 13 | oveq12d 7468 | . . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) |
15 | 12 | oveq1d 7465 | . . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ 𝑉) = ((𝐺‘𝑃) ∨ 𝑉)) |
16 | 14, 15 | eqeq12d 2756 | . . 3 ⊢ (𝐹 = 𝐺 → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉) ↔ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑉))) |
17 | vex 3492 | . . . . 5 ⊢ 𝑠 ∈ V | |
18 | eqid 2740 | . . . . . 6 ⊢ ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) | |
19 | 9, 18 | cdleme31sc 40343 | . . . . 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 2740 | . . . 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 2740 | . . . 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 2740 | . . . 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 | cdlemg2klem.v | . . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
25 | 1, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11, 24 | cdleme42keg 40445 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑉)) |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 25 | cdlemg2ce 40551 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉)) |
27 | 26 | 3com23 1126 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 Vcvv 3488 ⦋csb 3921 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6575 ℩crio 7405 (class class class)co 7450 Basecbs 17260 lecple 17320 joincjn 18383 meetcmee 18384 Atomscatm 39221 HLchlt 39308 LHypclh 39943 LTrncltrn 40060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-riotaBAD 38911 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-undef 8316 df-map 8888 df-proset 18367 df-poset 18385 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18504 df-clat 18571 df-oposet 39134 df-ol 39136 df-oml 39137 df-covers 39224 df-ats 39225 df-atl 39256 df-cvlat 39280 df-hlat 39309 df-llines 39457 df-lplanes 39458 df-lvols 39459 df-lines 39460 df-psubsp 39462 df-pmap 39463 df-padd 39755 df-lhyp 39947 df-laut 39948 df-ldil 40063 df-ltrn 40064 df-trl 40118 |
This theorem is referenced by: cdlemg2k 40560 |
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