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Theorem cdlemg1cex 39972
Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 39947? (Contributed by NM, 17-Apr-2013.)
Hypotheses
Ref Expression
cdlemg1c.l ≀ = (leβ€˜πΎ)
cdlemg1c.a 𝐴 = (Atomsβ€˜πΎ)
cdlemg1c.h 𝐻 = (LHypβ€˜πΎ)
cdlemg1c.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
cdlemg1cex ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐹 ∈ 𝑇 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))))
Distinct variable groups:   𝑓,𝑝,π‘ž,𝐴   𝑓,𝐹,𝑝,π‘ž   𝑓,𝐻,𝑝,π‘ž   𝑓,𝐾,𝑝,π‘ž   ≀ ,𝑓,𝑝,π‘ž   𝑇,𝑓,𝑝,π‘ž   𝑓,π‘Š,𝑝,π‘ž

Proof of Theorem cdlemg1cex
StepHypRef Expression
1 cdlemg1c.l . . . . . . . 8 ≀ = (leβ€˜πΎ)
2 cdlemg1c.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
3 cdlemg1c.h . . . . . . . 8 𝐻 = (LHypβ€˜πΎ)
4 cdlemg1c.t . . . . . . . 8 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4ltrnel 39523 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ ((πΉβ€˜π‘) ∈ 𝐴 ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š))
653expa 1115 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ ((πΉβ€˜π‘) ∈ 𝐴 ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š))
76simpld 494 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ (πΉβ€˜π‘) ∈ 𝐴)
8 simprr 770 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ Β¬ 𝑝 ≀ π‘Š)
96simprd 495 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ Β¬ (πΉβ€˜π‘) ≀ π‘Š)
10 simpll 764 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
11 simpr 484 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š))
12 simplr 766 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ 𝐹 ∈ 𝑇)
131, 2, 3, 4cdlemeiota 39969 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))
1410, 11, 12, 13syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))
15 breq1 5144 . . . . . . . 8 (π‘ž = (πΉβ€˜π‘) β†’ (π‘ž ≀ π‘Š ↔ (πΉβ€˜π‘) ≀ π‘Š))
1615notbid 318 . . . . . . 7 (π‘ž = (πΉβ€˜π‘) β†’ (Β¬ π‘ž ≀ π‘Š ↔ Β¬ (πΉβ€˜π‘) ≀ π‘Š))
17 eqeq2 2738 . . . . . . . . 9 (π‘ž = (πΉβ€˜π‘) β†’ ((π‘“β€˜π‘) = π‘ž ↔ (π‘“β€˜π‘) = (πΉβ€˜π‘)))
1817riotabidv 7363 . . . . . . . 8 (π‘ž = (πΉβ€˜π‘) β†’ (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž) = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))
1918eqeq2d 2737 . . . . . . 7 (π‘ž = (πΉβ€˜π‘) β†’ (𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž) ↔ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘))))
2016, 193anbi23d 1435 . . . . . 6 (π‘ž = (πΉβ€˜π‘) β†’ ((Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) ↔ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))))
2120rspcev 3606 . . . . 5 (((πΉβ€˜π‘) ∈ 𝐴 ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ (πΉβ€˜π‘) ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = (πΉβ€˜π‘)))) β†’ βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)))
227, 8, 9, 14, 21syl13anc 1369 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š)) β†’ βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)))
231, 2, 3lhpexnle 39390 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝 ≀ π‘Š)
2423adantr 480 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝 ≀ π‘Š)
2522, 24reximddv 3165 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)))
2625ex 412 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐹 ∈ 𝑇 β†’ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))))
27 simp1 1133 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
28 simp2l 1196 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ 𝑝 ∈ 𝐴)
29 simp31 1206 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ Β¬ 𝑝 ≀ π‘Š)
3028, 29jca 511 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š))
31 simp2r 1197 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ π‘ž ∈ 𝐴)
32 simp32 1207 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ Β¬ π‘ž ≀ π‘Š)
3331, 32jca 511 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ (π‘ž ∈ 𝐴 ∧ Β¬ π‘ž ≀ π‘Š))
34 simp33 1208 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))
351, 2, 3, 4cdlemg1ci2 39970 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š) ∧ (π‘ž ∈ 𝐴 ∧ Β¬ π‘ž ≀ π‘Š)) ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) β†’ 𝐹 ∈ 𝑇)
3627, 30, 33, 34, 35syl31anc 1370 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) ∧ (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))) β†’ 𝐹 ∈ 𝑇)
37363exp 1116 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((𝑝 ∈ 𝐴 ∧ π‘ž ∈ 𝐴) β†’ ((Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) β†’ 𝐹 ∈ 𝑇)))
3837rexlimdvv 3204 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž)) β†’ 𝐹 ∈ 𝑇))
3926, 38impbid 211 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐹 ∈ 𝑇 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š ∧ Β¬ π‘ž ≀ π‘Š ∧ 𝐹 = (℩𝑓 ∈ 𝑇 (π‘“β€˜π‘) = π‘ž))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6537  β„©crio 7360  lecple 17213  Atomscatm 38646  HLchlt 38733  LHypclh 39368  LTrncltrn 39485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-riotaBAD 38336
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-undef 8259  df-map 8824  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-llines 38882  df-lplanes 38883  df-lvols 38884  df-lines 38885  df-psubsp 38887  df-pmap 38888  df-padd 39180  df-lhyp 39372  df-laut 39373  df-ldil 39488  df-ltrn 39489  df-trl 39543
This theorem is referenced by:  cdlemg2cex  39975
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