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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemkuel | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma1 (p) function to be a translation. TODO: combine cdlemkj 38489? (Contributed by NM, 2-Jul-2013.) |
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 |
---|---|
cdlemkuel | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑈‘𝐺) ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp13 1206 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐺 ∈ 𝑇) | |
2 | cdlemk1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | cdlemk1.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | cdlemk1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | cdlemk1.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemk1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemk1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | cdlemk1.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
9 | cdlemk1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
10 | cdlemk1.u | . . . 4 ⊢ 𝑈 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷)))))) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdlemksv 38470 | . . 3 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷)))))) |
12 | 1, 11 | syl 17 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑈‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷)))))) |
13 | cdlemk1.s | . . 3 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹)))))) | |
14 | cdlemk1.o | . . 3 ⊢ 𝑂 = (𝑆‘𝐷) | |
15 | eqid 2738 | . . 3 ⊢ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))) | |
16 | 2, 3, 4, 9, 5, 6, 7, 8, 13, 14, 15 | cdlemkj 38489 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))) ∈ 𝑇) |
17 | 12, 16 | eqeltrd 2833 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑈‘𝐺) ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 class class class wbr 5027 ↦ cmpt 5107 I cid 5424 ◡ccnv 5518 ↾ cres 5521 ∘ ccom 5523 ‘cfv 6333 ℩crio 7120 (class class class)co 7164 Basecbs 16579 lecple 16668 joincjn 17663 meetcmee 17664 Atomscatm 36889 HLchlt 36976 LHypclh 37610 LTrncltrn 37727 trLctrl 37784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-riotaBAD 36579 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-undef 7961 df-map 8432 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 |
This theorem is referenced by: cdlemkuat 38492 cdlemkuv2 38493 cdlemk19 38495 cdlemk12u 38498 cdlemkuel-2N 38510 cdlemkuel-3 38524 |
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