cdlemg2ce - Mathbox for Norm Megill

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Theorem cdlemg2ce 41852

Description: Utility theorem to eliminate p,q when converting theorems with explicit f. TODO: fix comment. (Contributed by NM, 22-Apr-2013.)

**Hypotheses**
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))
cdlemg2ce.p	⊢ (𝐹 = 𝐺 → (𝜓 ↔ 𝜒))
cdlemg2ce.c	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ 𝜑) → 𝜒)

**Assertion**
Ref	Expression
cdlemg2ce	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) → 𝜓)

Distinct variable groups: ∧ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝐾,𝑠,𝑡,𝑥,𝑦,𝑧 𝑈,𝑠,𝑡,𝑥,𝑦,𝑧 𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝐵,𝑠,𝑡,𝑥,𝑦,𝑧 𝐴,𝑠,𝑡,𝑥,𝑦,𝑧 ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝐻,𝑠,𝑡,𝑥,𝑦,𝑧 𝑥,𝐸,𝑦,𝑧 𝐷,𝑠,𝑥,𝑦,𝑧 𝑞,𝑝,𝐴 𝐹,𝑝,𝑞 𝐻,𝑝,𝑞 𝐾,𝑝,𝑞 ≤ ,𝑝,𝑞 𝑇,𝑝,𝑞 𝑊,𝑝,𝑞,𝑠,𝑡,𝑥,𝑦,𝑧 𝜑,𝑝,𝑞 𝜓,𝑝,𝑞 Allowed substitution hints: 𝜑(𝑥, 𝑦, 𝑧, 𝑡, 𝑠) 𝜓(𝑥, 𝑦, 𝑧, 𝑡, 𝑠) 𝜒(𝑥, 𝑦, 𝑧, 𝑡, 𝑠, 𝑞, 𝑝) 𝐵(𝑞, 𝑝) 𝐷(𝑡, 𝑞, 𝑝) 𝑇(𝑥, 𝑦, 𝑧, 𝑡, 𝑠) 𝑈(𝑞, 𝑝) 𝐸(𝑡, 𝑠, 𝑞, 𝑝) 𝐹(𝑥, 𝑦, 𝑧, 𝑡, 𝑠) 𝐺(𝑥, 𝑦, 𝑧, 𝑡, 𝑠, 𝑞, 𝑝) ∨ (𝑞, 𝑝) ∧ (𝑞, 𝑝)

**Proof of Theorem cdlemg2ce**
Step	Hyp	Ref	Expression
1		simp2 1155	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) → 𝐹 ∈ 𝑇)
2		cdlemg2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		cdlemg2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
4		cdlemg2.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		cdlemg2.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
6		cdlemg2.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemg2.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemg2.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9		cdlemg2ex.u	. . . . 5 ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊)
10		cdlemg2ex.d	. . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))
11		cdlemg2ex.e	. . . . 5 ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
12		cdlemg2ex.g	. . . . 5 ⊢ 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))
13	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdlemg2cex 41851	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)))
14	13	3ad2ant1 1151	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)))
15	1, 14	mpbid 235	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺))
16		simp11 1222	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		simp2l 1218	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → 𝑝 ∈ 𝐴)
18		simp31 1228	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → ¬ 𝑝 ≤ 𝑊)
19	17, 18	jca 521	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊))
20		simp2r 1219	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → 𝑞 ∈ 𝐴)
21		simp32 1229	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → ¬ 𝑞 ≤ 𝑊)
22	20, 21	jca 521	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊))
23		simp13 1224	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → 𝜑)
24		cdlemg2ce.c	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ 𝜑) → 𝜒)
25	16, 19, 22, 23, 24	syl31anc 1401	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → 𝜒)
26		simp33 1230	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → 𝐹 = 𝐺)
27		cdlemg2ce.p	. . . . . 6 ⊢ (𝐹 = 𝐺 → (𝜓 ↔ 𝜒))
28	26, 27	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → (𝜓 ↔ 𝜒))
29	25, 28	mpbird 260	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)) → 𝜓)
30	29	3exp 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) → ((𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺) → 𝜓)))
31	30	rexlimdvv 3222	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺) → 𝜓))
32	15, 31	mpd 16	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝜑) → 𝜓)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 ⦋csb 3851 ifcif 4485 class class class wbr 5106 ↦ cmpt 5190 ‘cfv 6561 ℩crio 7402 (class class class)co 7446 Basecbs 17419 lecple 17467 joincjn 18517 meetcmee 18518 Atomscatm 40523 HLchlt 40610 LHypclh 41244 LTrncltrn 41361

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-riotaBAD 40213

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-1st 8028 df-2nd 8029 df-undef 8321 df-map 8880 df-proset 18500 df-poset 18519 df-plt 18534 df-lub 18550 df-glb 18551 df-join 18552 df-meet 18553 df-p0 18629 df-p1 18630 df-lat 18638 df-clat 18705 df-oposet 40436 df-ol 40438 df-oml 40439 df-covers 40526 df-ats 40527 df-atl 40558 df-cvlat 40582 df-hlat 40611 df-llines 40758 df-lplanes 40759 df-lvols 40760 df-lines 40761 df-psubsp 40763 df-pmap 40764 df-padd 41056 df-lhyp 41248 df-laut 41249 df-ldil 41364 df-ltrn 41365 df-trl 41419

This theorem is used by: cdlemg2jlemOLDN 41853 cdlemg2fvlem 41854 cdlemg2klem 41855

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