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Theorem ltexpri 6586
Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
Assertion
Ref Expression
ltexpri (A<P Bx P (A +P x) = B)
Distinct variable groups:   x,A   x,B

Proof of Theorem ltexpri
Dummy variables y z u v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 103 . . . . . . . 8 ((y = u z = v) → z = v)
21eleq1d 2103 . . . . . . 7 ((y = u z = v) → (z (2ndA) ↔ v (2ndA)))
3 simpl 102 . . . . . . . . 9 ((y = u z = v) → y = u)
41, 3oveq12d 5473 . . . . . . . 8 ((y = u z = v) → (z +Q y) = (v +Q u))
54eleq1d 2103 . . . . . . 7 ((y = u z = v) → ((z +Q y) (1stB) ↔ (v +Q u) (1stB)))
62, 5anbi12d 442 . . . . . 6 ((y = u z = v) → ((z (2ndA) (z +Q y) (1stB)) ↔ (v (2ndA) (v +Q u) (1stB))))
76cbvexdva 1801 . . . . 5 (y = u → (z(z (2ndA) (z +Q y) (1stB)) ↔ v(v (2ndA) (v +Q u) (1stB))))
87cbvrabv 2550 . . . 4 {y Qz(z (2ndA) (z +Q y) (1stB))} = {u Qv(v (2ndA) (v +Q u) (1stB))}
91eleq1d 2103 . . . . . . 7 ((y = u z = v) → (z (1stA) ↔ v (1stA)))
104eleq1d 2103 . . . . . . 7 ((y = u z = v) → ((z +Q y) (2ndB) ↔ (v +Q u) (2ndB)))
119, 10anbi12d 442 . . . . . 6 ((y = u z = v) → ((z (1stA) (z +Q y) (2ndB)) ↔ (v (1stA) (v +Q u) (2ndB))))
1211cbvexdva 1801 . . . . 5 (y = u → (z(z (1stA) (z +Q y) (2ndB)) ↔ v(v (1stA) (v +Q u) (2ndB))))
1312cbvrabv 2550 . . . 4 {y Qz(z (1stA) (z +Q y) (2ndB))} = {u Qv(v (1stA) (v +Q u) (2ndB))}
148, 13opeq12i 3545 . . 3 ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ = ⟨{u Qv(v (2ndA) (v +Q u) (1stB))}, {u Qv(v (1stA) (v +Q u) (2ndB))}⟩
1514ltexprlempr 6581 . 2 (A<P B → ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ P)
1614ltexprlemfl 6582 . . . 4 (A<P B → (1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) ⊆ (1stB))
1714ltexprlemrl 6583 . . . 4 (A<P B → (1stB) ⊆ (1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)))
1816, 17eqssd 2956 . . 3 (A<P B → (1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (1stB))
1914ltexprlemfu 6584 . . . 4 (A<P B → (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) ⊆ (2ndB))
2014ltexprlemru 6585 . . . 4 (A<P B → (2ndB) ⊆ (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)))
2119, 20eqssd 2956 . . 3 (A<P B → (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (2ndB))
22 ltrelpr 6487 . . . . . . 7 <P ⊆ (P × P)
2322brel 4335 . . . . . 6 (A<P B → (A P B P))
2423simpld 105 . . . . 5 (A<P BA P)
25 addclpr 6519 . . . . 5 ((A P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ P) → (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) P)
2624, 15, 25syl2anc 391 . . . 4 (A<P B → (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) P)
2723simprd 107 . . . 4 (A<P BB P)
28 preqlu 6454 . . . 4 (((A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) P B P) → ((A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B ↔ ((1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (1stB) (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (2ndB))))
2926, 27, 28syl2anc 391 . . 3 (A<P B → ((A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B ↔ ((1st ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (1stB) (2nd ‘(A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩)) = (2ndB))))
3018, 21, 29mpbir2and 850 . 2 (A<P B → (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B)
31 oveq2 5463 . . . 4 (x = ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ → (A +P x) = (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩))
3231eqeq1d 2045 . . 3 (x = ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ → ((A +P x) = B ↔ (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B))
3332rspcev 2650 . 2 ((⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩ P (A +P ⟨{y Qz(z (2ndA) (z +Q y) (1stB))}, {y Qz(z (1stA) (z +Q y) (2ndB))}⟩) = B) → x P (A +P x) = B)
3415, 30, 33syl2anc 391 1 (A<P Bx P (A +P x) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wrex 2301  {crab 2304  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  Pcnp 6275   +P cpp 6277  <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-iplp 6450  df-iltp 6452
This theorem is referenced by:  ltaprlem  6590  ltaprg  6591  ltmprr  6613  recexgt0sr  6681  mulgt0sr  6684
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