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Theorem ltexpri 6654
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 𝐵 → ∃𝑥P (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 = 𝑢𝑧 = 𝑣) → 𝑧 = 𝑣)
21eleq1d 2106 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 ∈ (2nd𝐴) ↔ 𝑣 ∈ (2nd𝐴)))
3 simpl 102 . . . . . . . . 9 ((𝑦 = 𝑢𝑧 = 𝑣) → 𝑦 = 𝑢)
41, 3oveq12d 5491 . . . . . . . 8 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 +Q 𝑦) = (𝑣 +Q 𝑢))
54eleq1d 2106 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 +Q 𝑦) ∈ (1st𝐵) ↔ (𝑣 +Q 𝑢) ∈ (1st𝐵)))
62, 5anbi12d 442 . . . . . 6 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵)) ↔ (𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))))
76cbvexdva 1804 . . . . 5 (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵)) ↔ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))))
87cbvrabv 2553 . . . 4 {𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))} = {𝑢Q ∣ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))}
91eleq1d 2106 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → (𝑧 ∈ (1st𝐴) ↔ 𝑣 ∈ (1st𝐴)))
104eleq1d 2106 . . . . . . 7 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 +Q 𝑦) ∈ (2nd𝐵) ↔ (𝑣 +Q 𝑢) ∈ (2nd𝐵)))
119, 10anbi12d 442 . . . . . 6 ((𝑦 = 𝑢𝑧 = 𝑣) → ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵)) ↔ (𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))))
1211cbvexdva 1804 . . . . 5 (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵)) ↔ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))))
1312cbvrabv 2553 . . . 4 {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))} = {𝑢Q ∣ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))}
148, 13opeq12i 3550 . . 3 ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ = ⟨{𝑢Q ∣ ∃𝑣(𝑣 ∈ (2nd𝐴) ∧ (𝑣 +Q 𝑢) ∈ (1st𝐵))}, {𝑢Q ∣ ∃𝑣(𝑣 ∈ (1st𝐴) ∧ (𝑣 +Q 𝑢) ∈ (2nd𝐵))}⟩
1514ltexprlempr 6649 . 2 (𝐴<P 𝐵 → ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P)
1614ltexprlemfl 6650 . . . 4 (𝐴<P 𝐵 → (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) ⊆ (1st𝐵))
1714ltexprlemrl 6651 . . . 4 (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)))
1816, 17eqssd 2959 . . 3 (𝐴<P 𝐵 → (1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵))
1914ltexprlemfu 6652 . . . 4 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) ⊆ (2nd𝐵))
2014ltexprlemru 6653 . . . 4 (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)))
2119, 20eqssd 2959 . . 3 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))
22 ltrelpr 6546 . . . . . . 7 <P ⊆ (P × P)
2322brel 4353 . . . . . 6 (𝐴<P 𝐵 → (𝐴P𝐵P))
2423simpld 105 . . . . 5 (𝐴<P 𝐵𝐴P)
25 addclpr 6578 . . . . 5 ((𝐴P ∧ ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P) → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P)
2624, 15, 25syl2anc 391 . . . 4 (𝐴<P 𝐵 → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P)
2723simprd 107 . . . 4 (𝐴<P 𝐵𝐵P)
28 preqlu 6513 . . . 4 (((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) ∈ P𝐵P) → ((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵 ↔ ((1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵) ∧ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))))
2926, 27, 28syl2anc 391 . . 3 (𝐴<P 𝐵 → ((𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵 ↔ ((1st ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (1st𝐵) ∧ (2nd ‘(𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩)) = (2nd𝐵))))
3018, 21, 29mpbir2and 851 . 2 (𝐴<P 𝐵 → (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵)
31 oveq2 5481 . . . 4 (𝑥 = ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ → (𝐴 +P 𝑥) = (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩))
3231eqeq1d 2048 . . 3 (𝑥 = ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵))
3332rspcev 2653 . 2 ((⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩ ∈ P ∧ (𝐴 +P ⟨{𝑦Q ∣ ∃𝑧(𝑧 ∈ (2nd𝐴) ∧ (𝑧 +Q 𝑦) ∈ (1st𝐵))}, {𝑦Q ∣ ∃𝑧(𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑦) ∈ (2nd𝐵))}⟩) = 𝐵) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
3415, 30, 33syl2anc 391 1 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393  wrex 2304  {crab 2307  cop 3375   class class class wbr 3760  cfv 4863  (class class class)co 5473  1st c1st 5726  2nd c2nd 5727  Qcnq 6321   +Q cplq 6323  Pcnp 6332   +P cpp 6334  <P cltp 6336
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3868  ax-sep 3871  ax-nul 3879  ax-pow 3923  ax-pr 3940  ax-un 4141  ax-setind 4230  ax-iinf 4272
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-int 3612  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-tr 3851  df-eprel 4022  df-id 4026  df-po 4029  df-iso 4030  df-iord 4074  df-on 4076  df-suc 4079  df-iom 4275  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-f 4867  df-f1 4868  df-fo 4869  df-f1o 4870  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-1st 5728  df-2nd 5729  df-recs 5881  df-irdg 5918  df-1o 5962  df-2o 5963  df-oadd 5966  df-omul 5967  df-er 6065  df-ec 6067  df-qs 6071  df-ni 6345  df-pli 6346  df-mi 6347  df-lti 6348  df-plpq 6385  df-mpq 6386  df-enq 6388  df-nqqs 6389  df-plqqs 6390  df-mqqs 6391  df-1nqqs 6392  df-rq 6393  df-ltnqqs 6394  df-enq0 6465  df-nq0 6466  df-0nq0 6467  df-plq0 6468  df-mq0 6469  df-inp 6507  df-iplp 6509  df-iltp 6511
This theorem is referenced by:  lteupri  6658  ltaprlem  6659  ltaprg  6660  ltmprr  6683  recexgt0sr  6801  mulgt0sr  6805
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