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Theorem aptiprleml 6609
Description: Lemma for aptipr 6611. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptiprleml ((A P B P ¬ B<P A) → (1stA) ⊆ (1stB))

Proof of Theorem aptiprleml
Dummy variables f g 𝑠 𝑡 u v x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6457 . . . . . . 7 (A P → ⟨(1stA), (2ndA)⟩ P)
2 prnmaxl 6470 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P x (1stA)) → 𝑠 (1stA)x <Q 𝑠)
31, 2sylan 267 . . . . . 6 ((A P x (1stA)) → 𝑠 (1stA)x <Q 𝑠)
43ad2ant2rl 480 . . . . 5 (((A P B P) B<P A x (1stA))) → 𝑠 (1stA)x <Q 𝑠)
5 ltexnqi 6392 . . . . . . 7 (x <Q 𝑠𝑡 Q (x +Q 𝑡) = 𝑠)
65ad2antll 460 . . . . . 6 ((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) → 𝑡 Q (x +Q 𝑡) = 𝑠)
7 simplr 482 . . . . . . . . 9 (((A P B P) B<P A x (1stA))) → B P)
87ad2antrr 457 . . . . . . . 8 (((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) → B P)
9 simprl 483 . . . . . . . 8 (((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) → 𝑡 Q)
10 prop 6457 . . . . . . . . 9 (B P → ⟨(1stB), (2ndB)⟩ P)
11 prarloc2 6486 . . . . . . . . 9 ((⟨(1stB), (2ndB)⟩ P 𝑡 Q) → u (1stB)(u +Q 𝑡) (2ndB))
1210, 11sylan 267 . . . . . . . 8 ((B P 𝑡 Q) → u (1stB)(u +Q 𝑡) (2ndB))
138, 9, 12syl2anc 391 . . . . . . 7 (((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) → u (1stB)(u +Q 𝑡) (2ndB))
148adantr 261 . . . . . . . . . 10 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → B P)
15 simprl 483 . . . . . . . . . 10 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → u (1stB))
16 elprnql 6463 . . . . . . . . . . 11 ((⟨(1stB), (2ndB)⟩ P u (1stB)) → u Q)
1710, 16sylan 267 . . . . . . . . . 10 ((B P u (1stB)) → u Q)
1814, 15, 17syl2anc 391 . . . . . . . . 9 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → u Q)
19 simpll 481 . . . . . . . . . . 11 (((A P B P) B<P A x (1stA))) → A P)
2019ad3antrrr 461 . . . . . . . . . 10 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → A P)
21 simprr 484 . . . . . . . . . . 11 (((A P B P) B<P A x (1stA))) → x (1stA))
2221ad3antrrr 461 . . . . . . . . . 10 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → x (1stA))
23 elprnql 6463 . . . . . . . . . . 11 ((⟨(1stA), (2ndA)⟩ P x (1stA)) → x Q)
241, 23sylan 267 . . . . . . . . . 10 ((A P x (1stA)) → x Q)
2520, 22, 24syl2anc 391 . . . . . . . . 9 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → x Q)
26 nqtri3or 6380 . . . . . . . . 9 ((u Q x Q) → (u <Q x u = x x <Q u))
2718, 25, 26syl2anc 391 . . . . . . . 8 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (u <Q x u = x x <Q u))
2818adantr 261 . . . . . . . . . . . . . 14 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → u Q)
29 simplrl 487 . . . . . . . . . . . . . . 15 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → 𝑡 Q)
3029adantr 261 . . . . . . . . . . . . . 14 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → 𝑡 Q)
31 addclnq 6359 . . . . . . . . . . . . . 14 ((u Q 𝑡 Q) → (u +Q 𝑡) Q)
3228, 30, 31syl2anc 391 . . . . . . . . . . . . 13 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → (u +Q 𝑡) Q)
33 ltanqg 6384 . . . . . . . . . . . . . . . . . 18 ((f Q g Q Q) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
3433adantl 262 . . . . . . . . . . . . . . . . 17 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) (f Q g Q Q)) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
35 addcomnqg 6365 . . . . . . . . . . . . . . . . . 18 ((f Q g Q) → (f +Q g) = (g +Q f))
3635adantl 262 . . . . . . . . . . . . . . . . 17 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) (f Q g Q)) → (f +Q g) = (g +Q f))
3734, 18, 25, 29, 36caovord2d 5612 . . . . . . . . . . . . . . . 16 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (u <Q x ↔ (u +Q 𝑡) <Q (x +Q 𝑡)))
38 simplrr 488 . . . . . . . . . . . . . . . . . 18 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (x +Q 𝑡) = 𝑠)
39 simprl 483 . . . . . . . . . . . . . . . . . . 19 ((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) → 𝑠 (1stA))
4039ad2antrr 457 . . . . . . . . . . . . . . . . . 18 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → 𝑠 (1stA))
4138, 40eqeltrd 2111 . . . . . . . . . . . . . . . . 17 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (x +Q 𝑡) (1stA))
42 prcdnql 6466 . . . . . . . . . . . . . . . . . 18 ((⟨(1stA), (2ndA)⟩ P (x +Q 𝑡) (1stA)) → ((u +Q 𝑡) <Q (x +Q 𝑡) → (u +Q 𝑡) (1stA)))
431, 42sylan 267 . . . . . . . . . . . . . . . . 17 ((A P (x +Q 𝑡) (1stA)) → ((u +Q 𝑡) <Q (x +Q 𝑡) → (u +Q 𝑡) (1stA)))
4420, 41, 43syl2anc 391 . . . . . . . . . . . . . . . 16 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → ((u +Q 𝑡) <Q (x +Q 𝑡) → (u +Q 𝑡) (1stA)))
4537, 44sylbid 139 . . . . . . . . . . . . . . 15 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (u <Q x → (u +Q 𝑡) (1stA)))
46 simprr 484 . . . . . . . . . . . . . . 15 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (u +Q 𝑡) (2ndB))
4745, 46jctild 299 . . . . . . . . . . . . . 14 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (u <Q x → ((u +Q 𝑡) (2ndB) (u +Q 𝑡) (1stA))))
4847imp 115 . . . . . . . . . . . . 13 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → ((u +Q 𝑡) (2ndB) (u +Q 𝑡) (1stA)))
49 eleq1 2097 . . . . . . . . . . . . . . 15 (v = (u +Q 𝑡) → (v (2ndB) ↔ (u +Q 𝑡) (2ndB)))
50 eleq1 2097 . . . . . . . . . . . . . . 15 (v = (u +Q 𝑡) → (v (1stA) ↔ (u +Q 𝑡) (1stA)))
5149, 50anbi12d 442 . . . . . . . . . . . . . 14 (v = (u +Q 𝑡) → ((v (2ndB) v (1stA)) ↔ ((u +Q 𝑡) (2ndB) (u +Q 𝑡) (1stA))))
5251rspcev 2650 . . . . . . . . . . . . 13 (((u +Q 𝑡) Q ((u +Q 𝑡) (2ndB) (u +Q 𝑡) (1stA))) → v Q (v (2ndB) v (1stA)))
5332, 48, 52syl2anc 391 . . . . . . . . . . . 12 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → v Q (v (2ndB) v (1stA)))
54 ltdfpr 6488 . . . . . . . . . . . . . 14 ((B P A P) → (B<P Av Q (v (2ndB) v (1stA))))
5514, 20, 54syl2anc 391 . . . . . . . . . . . . 13 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (B<P Av Q (v (2ndB) v (1stA))))
5655adantr 261 . . . . . . . . . . . 12 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → (B<P Av Q (v (2ndB) v (1stA))))
5753, 56mpbird 156 . . . . . . . . . . 11 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → B<P A)
58 simplrl 487 . . . . . . . . . . . 12 ((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) → ¬ B<P A)
5958ad3antrrr 461 . . . . . . . . . . 11 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → ¬ B<P A)
6057, 59pm2.21dd 550 . . . . . . . . . 10 (((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) u <Q x) → x (1stB))
6160ex 108 . . . . . . . . 9 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (u <Q xx (1stB)))
62 eleq1 2097 . . . . . . . . . 10 (u = x → (u (1stB) ↔ x (1stB)))
6315, 62syl5ibcom 144 . . . . . . . . 9 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (u = xx (1stB)))
64 prcdnql 6466 . . . . . . . . . . 11 ((⟨(1stB), (2ndB)⟩ P u (1stB)) → (x <Q ux (1stB)))
6510, 64sylan 267 . . . . . . . . . 10 ((B P u (1stB)) → (x <Q ux (1stB)))
6614, 15, 65syl2anc 391 . . . . . . . . 9 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → (x <Q ux (1stB)))
6761, 63, 663jaod 1198 . . . . . . . 8 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → ((u <Q x u = x x <Q u) → x (1stB)))
6827, 67mpd 13 . . . . . . 7 ((((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) (u (1stB) (u +Q 𝑡) (2ndB))) → x (1stB))
6913, 68rexlimddv 2431 . . . . . 6 (((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) (𝑡 Q (x +Q 𝑡) = 𝑠)) → x (1stB))
706, 69rexlimddv 2431 . . . . 5 ((((A P B P) B<P A x (1stA))) (𝑠 (1stA) x <Q 𝑠)) → x (1stB))
714, 70rexlimddv 2431 . . . 4 (((A P B P) B<P A x (1stA))) → x (1stB))
7271expr 357 . . 3 (((A P B P) ¬ B<P A) → (x (1stA) → x (1stB)))
73723impa 1098 . 2 ((A P B P ¬ B<P A) → (x (1stA) → x (1stB)))
7473ssrdv 2945 1 ((A P B P ¬ B<P A) → (1stA) ⊆ (1stB))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   w3o 883   w3a 884   = wceq 1242   wcel 1390  wrex 2301  wss 2911  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iltp 6452
This theorem is referenced by:  aptipr  6611
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