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Theorem ltaprlem 9851
Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltaprlem (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Proof of Theorem ltaprlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ltrelpr 9805 . . . . . 6 <P ⊆ (P × P)
21brel 5158 . . . . 5 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simpld 475 . . . 4 (𝐴<P 𝐵𝐴P)
4 ltexpri 9850 . . . . 5 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
5 addclpr 9825 . . . . . . . 8 ((𝐶P𝐴P) → (𝐶 +P 𝐴) ∈ P)
6 ltaddpr 9841 . . . . . . . . . 10 (((𝐶 +P 𝐴) ∈ P𝑥P) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥))
7 addasspr 9829 . . . . . . . . . . . 12 ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥))
8 oveq2 6643 . . . . . . . . . . . 12 ((𝐴 +P 𝑥) = 𝐵 → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
97, 8syl5eq 2666 . . . . . . . . . . 11 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
109breq2d 4656 . . . . . . . . . 10 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
116, 10syl5ib 234 . . . . . . . . 9 ((𝐴 +P 𝑥) = 𝐵 → (((𝐶 +P 𝐴) ∈ P𝑥P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
1211expd 452 . . . . . . . 8 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴) ∈ P → (𝑥P → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
135, 12syl5 34 . . . . . . 7 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶P𝐴P) → (𝑥P → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1413com3r 87 . . . . . 6 (𝑥P → ((𝐴 +P 𝑥) = 𝐵 → ((𝐶P𝐴P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1514rexlimiv 3023 . . . . 5 (∃𝑥P (𝐴 +P 𝑥) = 𝐵 → ((𝐶P𝐴P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
164, 15syl 17 . . . 4 (𝐴<P 𝐵 → ((𝐶P𝐴P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
173, 16sylan2i 686 . . 3 (𝐴<P 𝐵 → ((𝐶P𝐴<P 𝐵) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
1817expd 452 . 2 (𝐴<P 𝐵 → (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1918pm2.43b 55 1 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wrex 2910   class class class wbr 4644  (class class class)co 6635  Pcnp 9666   +P cpp 9668  <P cltp 9670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-omul 7550  df-er 7727  df-ni 9679  df-pli 9680  df-mi 9681  df-lti 9682  df-plpq 9715  df-mpq 9716  df-ltpq 9717  df-enq 9718  df-nq 9719  df-erq 9720  df-plq 9721  df-mq 9722  df-1nq 9723  df-rq 9724  df-ltnq 9725  df-np 9788  df-plp 9790  df-ltp 9792
This theorem is referenced by:  ltapr  9852
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