cdlemg1a - Mathbox for Norm Megill

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Theorem cdlemg1a 40173

Description: Shorter expression for 𝐺. TODO: fix comment. TODO: shorten using cdleme 40163 or vice-versa? Also, if not shortened with cdleme 40163, then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-Apr-2013.)

**Hypotheses**
Ref	Expression
cdlemg1.b	⊢ 𝐵 = (Base‘𝐾)
cdlemg1.l	⊢ ≤ = (le‘𝐾)
cdlemg1.j	⊢ ∨ = (join‘𝐾)
cdlemg1.m	⊢ ∧ = (meet‘𝐾)
cdlemg1.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg1.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg1.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemg1.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemg1.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemg1.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))
cdlemg1.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemg1a	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))

Distinct variable groups: 𝑡,𝑠,𝑥,𝑦,𝑧,𝐴,𝑓 𝐵,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝐷,𝑓,𝑠,𝑥,𝑦,𝑧 𝑓,𝐸,𝑥,𝑦,𝑧 𝐻,𝑠,𝑡,𝑥,𝑦,𝑧 ∨ ,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝐾,𝑠,𝑡,𝑥,𝑦,𝑧 ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧 ∧ ,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝑃,𝑠,𝑡,𝑥,𝑦,𝑧 𝑄,𝑠,𝑡,𝑥,𝑦,𝑧 𝑈,𝑠,𝑡,𝑥,𝑦,𝑧 𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 𝐴,𝑓 𝑓,𝐻 𝑓,𝐾 ≤ ,𝑓 𝑃,𝑓 𝑄,𝑓 𝑇,𝑓 𝑓,𝑊 𝑓,𝐺 Allowed substitution hints: 𝐷(𝑡) 𝑇(𝑥,𝑦,𝑧,𝑡,𝑠) 𝑈(𝑓) 𝐸(𝑡,𝑠) 𝐺(𝑥,𝑦,𝑧,𝑡,𝑠)

**Proof of Theorem cdlemg1a**
Step	Hyp	Ref	Expression
1		cdlemg1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemg1.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		cdlemg1.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		cdlemg1.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		cdlemg1.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemg1.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemg1.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdlemg1.d	. . . 4 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
9		cdlemg1.e	. . . 4 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
10		cdlemg1.g	. . . 4 ⊢ 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))
11		cdlemg1.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cdleme50ltrn 40160	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
13		simpll1 1209	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		simplr 767	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 ∈ 𝑇)
15	12	ad2antrr 724	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝐺 ∈ 𝑇)
16		simpll2 1210	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
17		simpr 483	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑓‘𝑃) = 𝑄)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme17d 40101	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑄)
19	18	ad2antrr 724	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝐺‘𝑃) = 𝑄)
20	17, 19	eqtr4d 2768	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑓‘𝑃) = (𝐺‘𝑃))
21	2, 5, 6, 11	cdlemd 39810	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑓‘𝑃) = (𝐺‘𝑃)) → 𝑓 = 𝐺)
22	13, 14, 15, 16, 20, 21	syl311anc 1381	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 = 𝐺)
23	22	ex 411	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 → 𝑓 = 𝐺))
24	18	adantr 479	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝐺‘𝑃) = 𝑄)
25		fveq1 6895	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑃) = (𝐺‘𝑃))
26	25	eqeq1d 2727	. . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑃) = 𝑄 ↔ (𝐺‘𝑃) = 𝑄))
27	24, 26	syl5ibrcom 246	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝑓 = 𝐺 → (𝑓‘𝑃) = 𝑄))
28	23, 27	impbid 211	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 ↔ 𝑓 = 𝐺))
29	12, 28	riota5 7405	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) = 𝐺)
30	29	eqcomd 2731	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ⦋csb 3889 ifcif 4530 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 Basecbs 17183 lecple 17243 joincjn 18306 meetcmee 18307 Atomscatm 38865 HLchlt 38952 LHypclh 39587 LTrncltrn 39704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-riotaBAD 38555

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-undef 8279 df-map 8847 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762

This theorem is referenced by: cdlemg1b2 40174

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