Theorem cdlemg1a 41364

Description: Shorter expression for 𝐺. TODO: fix comment. TODO: shorten using cdleme 41354 or vice-versa? Also, if not shortened with cdleme 41354, 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 41351	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
13		simpll1 1231	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14		simplr 780	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 ∈ 𝑇)
15	12	ad2antrr 738	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝐺 ∈ 𝑇)
16		simpll2 1232	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
17		simpr 489	. . . . . . 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 41292	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑄)
19	18	ad2antrr 738	. . . . . . 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 2801	. . . . . 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 41001	. . . . . 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 1411	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 = 𝐺)
23	22	ex 417	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 → 𝑓 = 𝐺))
24	18	adantr 485	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝐺‘𝑃) = 𝑄)
25		fveq1 6880	. . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑃) = (𝐺‘𝑃))
26	25	eqeq1d 2765	. . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑃) = 𝑄 ↔ (𝐺‘𝑃) = 𝑄))
27	24, 26	syl5ibrcom 250	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝑓 = 𝐺 → (𝑓‘𝑃) = 𝑄))
28	23, 27	impbid 215	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 ↔ 𝑓 = 𝐺))
29	12, 28	riota5 7396	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) = 𝐺)
30	29	eqcomd 2769	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ⦋csb 3853  ifcif 4487   class class class wbr 5109   ↦ cmpt 5192  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  Basecbs 17264  lecple 17312  joincjn 18362  meetcmee 18363  Atomscatm 40057  HLchlt 40144  LHypclh 40778  LTrncltrn 40895

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 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-riotaBAD 39747

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 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-undef 8265  df-map 8822  df-proset 18345  df-poset 18364  df-plt 18379  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-p0 18474  df-p1 18475  df-lat 18483  df-clat 18550  df-oposet 39970  df-ol 39972  df-oml 39973  df-covers 40060  df-ats 40061  df-atl 40092  df-cvlat 40116  df-hlat 40145  df-llines 40292  df-lplanes 40293  df-lvols 40294  df-lines 40295  df-psubsp 40297  df-pmap 40298  df-padd 40590  df-lhyp 40782  df-laut 40783  df-ldil 40898  df-ltrn 40899  df-trl 40953

This theorem is referenced by:  cdlemg1b2  41365