Theorem cdlemg1cex 41754

Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 41729? (Contributed by NM, 17-Apr-2013.)

Hypotheses

Ref	Expression
cdlemg1c.l	⊢ ≤ = (le‘𝐾)
cdlemg1c.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemg1c.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemg1c.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemg1cex	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))))

Distinct variable groups:   𝑓,𝑝,𝑞,𝐴   𝑓,𝐹,𝑝,𝑞   𝑓,𝐻,𝑝,𝑞   𝑓,𝐾,𝑝,𝑞   ≤ ,𝑓,𝑝,𝑞   𝑇,𝑓,𝑝,𝑞   𝑓,𝑊,𝑝,𝑞

Proof of Theorem cdlemg1cex

Step	Hyp	Ref	Expression
1		cdlemg1c.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
2		cdlemg1c.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
3		cdlemg1c.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		cdlemg1c.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	1, 2, 3, 4	ltrnel 41305	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ((𝐹‘𝑝) ∈ 𝐴 ∧ ¬ (𝐹‘𝑝) ≤ 𝑊))
6	5	3expa 1136	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ((𝐹‘𝑝) ∈ 𝐴 ∧ ¬ (𝐹‘𝑝) ≤ 𝑊))
7	6	simpld 500	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝐹‘𝑝) ∈ 𝐴)
8		simprr 785	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ¬ 𝑝 ≤ 𝑊)
9	6	simprd 501	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ¬ (𝐹‘𝑝) ≤ 𝑊)
10		simpll 779	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simpr 490	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊))
12		simplr 781	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
13	1, 2, 3, 4	cdlemeiota 41751	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))
14	10, 11, 12, 13	syl3anc 1398	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))
15		breq1 5105	. . . . . . . 8 ⊢ (𝑞 = (𝐹‘𝑝) → (𝑞 ≤ 𝑊 ↔ (𝐹‘𝑝) ≤ 𝑊))
16	15	notbid 321	. . . . . . 7 ⊢ (𝑞 = (𝐹‘𝑝) → (¬ 𝑞 ≤ 𝑊 ↔ ¬ (𝐹‘𝑝) ≤ 𝑊))
17		eqeq2 2772	. . . . . . . . 9 ⊢ (𝑞 = (𝐹‘𝑝) → ((𝑓‘𝑝) = 𝑞 ↔ (𝑓‘𝑝) = (𝐹‘𝑝)))
18	17	riotabidv 7383	. . . . . . . 8 ⊢ (𝑞 = (𝐹‘𝑝) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))
19	18	eqeq2d 2771	. . . . . . 7 ⊢ (𝑞 = (𝐹‘𝑝) → (𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝))))
20	16, 19	3anbi23d 1467	. . . . . 6 ⊢ (𝑞 = (𝐹‘𝑝) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) ↔ (¬ 𝑝 ≤ 𝑊 ∧ ¬ (𝐹‘𝑝) ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))))
21	20	rspcev 3576	. . . . 5 ⊢ (((𝐹‘𝑝) ∈ 𝐴 ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ (𝐹‘𝑝) ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))) → ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)))
22	7, 8, 9, 14, 21	syl13anc 1399	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)))
23	1, 2, 3	lhpexnle 41172	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊)
24	23	adantr 486	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊)
25	22, 24	reximddv 3178	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)))
26	25	ex 418	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))))
27		simp1 1154	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		simp2l 1218	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → 𝑝 ∈ 𝐴)
29		simp31 1228	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → ¬ 𝑝 ≤ 𝑊)
30	28, 29	jca 521	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊))
31		simp2r 1219	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → 𝑞 ∈ 𝐴)
32		simp32 1229	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → ¬ 𝑞 ≤ 𝑊)
33	31, 32	jca 521	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊))
34		simp33 1230	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))
35	1, 2, 3, 4	cdlemg1ci2 41752	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) → 𝐹 ∈ 𝑇)
36	27, 30, 33, 34, 35	syl31anc 1400	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → 𝐹 ∈ 𝑇)
37	36	3exp 1137	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) → 𝐹 ∈ 𝑇)))
38	37	rexlimdvv 3218	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) → 𝐹 ∈ 𝑇))
39	26, 38	impbid 215	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  ‘cfv 6543  ℩crio 7380  lecple 17444  Atomscatm 40429  HLchlt 40516  LHypclh 41150  LTrncltrn 41267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-riotaBAD 40119

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-undef 8297  df-map 8856  df-proset 18477  df-poset 18496  df-plt 18511  df-lub 18527  df-glb 18528  df-join 18529  df-meet 18530  df-p0 18606  df-p1 18607  df-lat 18615  df-clat 18682  df-oposet 40342  df-ol 40344  df-oml 40345  df-covers 40432  df-ats 40433  df-atl 40464  df-cvlat 40488  df-hlat 40517  df-llines 40664  df-lplanes 40665  df-lvols 40666  df-lines 40667  df-psubsp 40669  df-pmap 40670  df-padd 40962  df-lhyp 41154  df-laut 41155  df-ldil 41270  df-ltrn 41271  df-trl 41325

This theorem is used by:  cdlemg2cex  41757