cdlemg1b2 - Mathbox for Norm Megill

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Theorem cdlemg1b2 38594

Description: This theorem can be used to shorten 𝐺 = hypothesis. TODO: Fix comment. (Contributed by NM, 18-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)
cdlemg1.f	⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)

**Assertion**
Ref	Expression
cdlemg1b2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 = 𝐺)

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) 𝐹(𝑥,𝑦,𝑧,𝑡,𝑓,𝑠) 𝐺(𝑥,𝑦,𝑧,𝑡,𝑠)

**Proof of Theorem cdlemg1b2**
Step	Hyp	Ref	Expression
1		cdlemg1.f	. . 3 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)
2		cdlemg1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		cdlemg1.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		cdlemg1.j	. . . 4 ⊢ ∨ = (join‘𝐾)
5		cdlemg1.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
6		cdlemg1.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemg1.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemg1.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
9		cdlemg1.d	. . . 4 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
10		cdlemg1.e	. . . 4 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
11		eqid 2740	. . . 4 ⊢ (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)) = (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		cdlemg1.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemg1a 38593	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (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)) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))
14	1, 13	eqtr4id 2799	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 = (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)))
15		cdlemg1.g	. 2 ⊢ 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))
16	14, 15	eqtr4di 2798	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 = 𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 ⦋csb 3837 ifcif 4465 class class class wbr 5079 ↦ cmpt 5162 ‘cfv 6432 ℩crio 7228 (class class class)co 7272 Basecbs 16923 lecple 16980 joincjn 18040 meetcmee 18041 Atomscatm 37286 HLchlt 37373 LHypclh 38007 LTrncltrn 38124

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 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-riotaBAD 36976

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-1st 7825 df-2nd 7826 df-undef 8081 df-map 8609 df-proset 18024 df-poset 18042 df-plt 18059 df-lub 18075 df-glb 18076 df-join 18077 df-meet 18078 df-p0 18154 df-p1 18155 df-lat 18161 df-clat 18228 df-oposet 37199 df-ol 37201 df-oml 37202 df-covers 37289 df-ats 37290 df-atl 37321 df-cvlat 37345 df-hlat 37374 df-llines 37521 df-lplanes 37522 df-lvols 37523 df-lines 37524 df-psubsp 37526 df-pmap 37527 df-padd 37819 df-lhyp 38011 df-laut 38012 df-ldil 38127 df-ltrn 38128 df-trl 38182

This theorem is referenced by: cdlemg1idlemN 38595 cdlemg1fvawlemN 38596 cdlemg1ltrnlem 38597 cdlemg1finvtrlemN 38598 cdlemg1bOLDN 38599 cdlemg2cN 38612 cdlemg2cex 38614

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