Theorem cdlemg2cex 41346

Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 41318? (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))

Assertion

Ref	Expression
cdlemg2cex	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)))

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

Allowed substitution hints:   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 cdlemg2cex

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemg2.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		cdlemg2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		cdlemg2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		cdlemg2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	1, 2, 3, 4	cdlemg1cex 41343	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))))
6		simplll 786	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝐾 ∈ HL)
7		simpllr 787	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑊 ∈ 𝐻)
8		simplrl 788	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑝 ∈ 𝐴)
9		simprl 782	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → ¬ 𝑝 ≤ 𝑊)
10		simplrr 789	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑞 ∈ 𝐴)
11		simprr 784	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → ¬ 𝑞 ≤ 𝑊)
12		cdlemg2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
13		cdlemg2.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
14		cdlemg2.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
15		cdlemg2ex.u	. . . . . . . 8 ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊)
16		cdlemg2ex.d	. . . . . . . 8 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))
17		cdlemg2ex.e	. . . . . . . 8 ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
18		cdlemg2ex.g	. . . . . . . 8 ⊢ 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))
19		eqid 2763	. . . . . . . 8 ⊢ (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)
20	12, 1, 13, 14, 2, 3, 15, 16, 17, 18, 4, 19	cdlemg1b2 41326	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) = 𝐺)
21	6, 7, 8, 9, 10, 11, 20	syl222anc 1413	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) = 𝐺)
22	21	eqeq2d 2774	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) ↔ 𝐹 = 𝐺))
23	22	pm5.32da 589	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) → (((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) ↔ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝐹 = 𝐺)))
24		df-3an 1105	. . . 4 ⊢ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) ↔ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)))
25		df-3an 1105	. . . 4 ⊢ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺) ↔ ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊) ∧ 𝐹 = 𝐺))
26	23, 24, 25	3bitr4g 317	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) ↔ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)))
27	26	2rexbidva 3228	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)))
28	5, 27	bitrd 282	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = 𝐺)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  ⦋csb 3854  ifcif 4488   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6538  ℩crio 7368  (class class class)co 7412  Basecbs 17270  lecple 17318  joincjn 18368  meetcmee 18369  Atomscatm 40018  HLchlt 40105  LHypclh 40739  LTrncltrn 40856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-riotaBAD 39708

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-undef 8270  df-map 8827  df-proset 18351  df-poset 18370  df-plt 18385  df-lub 18401  df-glb 18402  df-join 18403  df-meet 18404  df-p0 18480  df-p1 18481  df-lat 18489  df-clat 18556  df-oposet 39931  df-ol 39933  df-oml 39934  df-covers 40021  df-ats 40022  df-atl 40053  df-cvlat 40077  df-hlat 40106  df-llines 40253  df-lplanes 40254  df-lvols 40255  df-lines 40256  df-psubsp 40258  df-pmap 40259  df-padd 40551  df-lhyp 40743  df-laut 40744  df-ldil 40859  df-ltrn 40860  df-trl 40914

This theorem is referenced by:  cdlemg2ce  41347