cdlemg1a - Mathbox for Norm Megill

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Theorem cdlemg1a 40537

Description: Shorter expression for 𝐺. TODO: fix comment. TODO: shorten using cdleme 40527 or vice-versa? Also, if not shortened with cdleme 40527, 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: ∧ ,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝑓,𝐸,𝑥,𝑦,𝑧 𝑈,𝑠,𝑡,𝑥,𝑦,𝑧 𝑓,𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 ∨ ,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝐵,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝑇,𝑓 𝐴,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 ≤ ,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝑓,𝐾,𝑠,𝑡,𝑥,𝑦,𝑧 𝑃,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝑄,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧 𝑓,𝐻,𝑠,𝑡,𝑥,𝑦,𝑧 𝐷,𝑓,𝑠,𝑥,𝑦,𝑧 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 40524	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
13		simpll1 1213	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		simplr 768	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 ∈ 𝑇)
15	12	ad2antrr 726	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝐺 ∈ 𝑇)
16		simpll2 1214	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
17		simpr 484	. . . . . . 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 40465	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑄)
19	18	ad2antrr 726	. . . . . . 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 2767	. . . . . 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 40174	. . . . . 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 1386	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 = 𝐺)
23	22	ex 412	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 → 𝑓 = 𝐺))
24	18	adantr 480	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝐺‘𝑃) = 𝑄)
25		fveq1 6839	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑃) = (𝐺‘𝑃))
26	25	eqeq1d 2731	. . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑃) = 𝑄 ↔ (𝐺‘𝑃) = 𝑄))
27	24, 26	syl5ibrcom 247	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝑓 = 𝐺 → (𝑓‘𝑃) = 𝑄))
28	23, 27	impbid 212	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 ↔ 𝑓 = 𝐺))
29	12, 28	riota5 7355	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) = 𝐺)
30	29	eqcomd 2735	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ⦋csb 3859 ifcif 4484 class class class wbr 5102 ↦ cmpt 5183 ‘cfv 6499 ℩crio 7325 (class class class)co 7369 Basecbs 17155 lecple 17203 joincjn 18248 meetcmee 18249 Atomscatm 39229 HLchlt 39316 LHypclh 39951 LTrncltrn 40068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-riotaBAD 38919

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-1st 7947 df-2nd 7948 df-undef 8229 df-map 8778 df-proset 18231 df-poset 18250 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-oposet 39142 df-ol 39144 df-oml 39145 df-covers 39232 df-ats 39233 df-atl 39264 df-cvlat 39288 df-hlat 39317 df-llines 39465 df-lplanes 39466 df-lvols 39467 df-lines 39468 df-psubsp 39470 df-pmap 39471 df-padd 39763 df-lhyp 39955 df-laut 39956 df-ldil 40071 df-ltrn 40072 df-trl 40126

This theorem is referenced by: cdlemg1b2 40538

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