Theorem cdlemg1cex 41080

Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 41055? (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 40631	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ((𝐹‘𝑝) ∈ 𝐴 ∧ ¬ (𝐹‘𝑝) ≤ 𝑊))
6	5	3expa 1124	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ((𝐹‘𝑝) ∈ 𝐴 ∧ ¬ (𝐹‘𝑝) ≤ 𝑊))
7	6	simpld 495	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝐹‘𝑝) ∈ 𝐴)
8		simprr 778	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ¬ 𝑝 ≤ 𝑊)
9	6	simprd 496	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → ¬ (𝐹‘𝑝) ≤ 𝑊)
10		simpll 772	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simpr 485	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊))
12		simplr 774	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
13	1, 2, 3, 4	cdlemeiota 41077	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))
14	10, 11, 12, 13	syl3anc 1379	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))
15		breq1 5075	. . . . . . . 8 ⊢ (𝑞 = (𝐹‘𝑝) → (𝑞 ≤ 𝑊 ↔ (𝐹‘𝑝) ≤ 𝑊))
16	15	notbid 319	. . . . . . 7 ⊢ (𝑞 = (𝐹‘𝑝) → (¬ 𝑞 ≤ 𝑊 ↔ ¬ (𝐹‘𝑝) ≤ 𝑊))
17		eqeq2 2751	. . . . . . . . 9 ⊢ (𝑞 = (𝐹‘𝑝) → ((𝑓‘𝑝) = 𝑞 ↔ (𝑓‘𝑝) = (𝐹‘𝑝)))
18	17	riotabidv 7315	. . . . . . . 8 ⊢ (𝑞 = (𝐹‘𝑝) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝)))
19	18	eqeq2d 2750	. . . . . . 7 ⊢ (𝑞 = (𝐹‘𝑝) → (𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞) ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = (𝐹‘𝑝))))
20	16, 19	3anbi23d 1447	. . . . . 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 3560	. . . . 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 1380	. . . 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 40498	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊)
24	23	adantr 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ∃𝑝 ∈ 𝐴 ¬ 𝑝 ≤ 𝑊)
25	22, 24	reximddv 3155	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)))
26	25	ex 413	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))))
27		simp1 1142	. . . . 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 1206	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → 𝑝 ∈ 𝐴)
29		simp31 1216	. . . . . 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 516	. . . . 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 1207	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → 𝑞 ∈ 𝐴)
32		simp32 1217	. . . . . 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 516	. . . . 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 1218	. . . . 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 41078	. . . . 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 1381	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) ∧ (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))) → 𝐹 ∈ 𝑇)
37	36	3exp 1125	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞)) → 𝐹 ∈ 𝑇)))
38	37	rexlimdvv 3195	. 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 213	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝑇 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑞 ≤ 𝑊 ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑝) = 𝑞))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   class class class wbr 5072  ‘cfv 6485  ℩crio 7312  lecple 17218  Atomscatm 39755  HLchlt 39842  LHypclh 40476  LTrncltrn 40593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-riotaBAD 39445

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-undef 8213  df-map 8765  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-llines 39990  df-lplanes 39991  df-lvols 39992  df-lines 39993  df-psubsp 39995  df-pmap 39996  df-padd 40288  df-lhyp 40480  df-laut 40481  df-ldil 40596  df-ltrn 40597  df-trl 40651

This theorem is referenced by:  cdlemg2cex  41083