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Theorem ltaprlem 9625
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 9579 . . . . . 6 <P ⊆ (P × P)
21brel 4984 . . . . 5 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simpld 473 . . . 4 (𝐴<P 𝐵𝐴P)
4 ltexpri 9624 . . . . 5 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
5 addclpr 9599 . . . . . . . 8 ((𝐶P𝐴P) → (𝐶 +P 𝐴) ∈ P)
6 ltaddpr 9615 . . . . . . . . . 10 (((𝐶 +P 𝐴) ∈ P𝑥P) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥))
7 addasspr 9603 . . . . . . . . . . . 12 ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥))
8 oveq2 6439 . . . . . . . . . . . 12 ((𝐴 +P 𝑥) = 𝐵 → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
97, 8syl5eq 2560 . . . . . . . . . . 11 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
109breq2d 4493 . . . . . . . . . 10 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
116, 10syl5ib 232 . . . . . . . . 9 ((𝐴 +P 𝑥) = 𝐵 → (((𝐶 +P 𝐴) ∈ P𝑥P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
1211expd 450 . . . . . . . 8 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶 +P 𝐴) ∈ P → (𝑥P → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
135, 12syl5 33 . . . . . . 7 ((𝐴 +P 𝑥) = 𝐵 → ((𝐶P𝐴P) → (𝑥P → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1413com3r 84 . . . . . 6 (𝑥P → ((𝐴 +P 𝑥) = 𝐵 → ((𝐶P𝐴P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1514rexlimiv 2913 . . . . 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 684 . . 3 (𝐴<P 𝐵 → ((𝐶P𝐴<P 𝐵) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
1817expd 450 . 2 (𝐴<P 𝐵 → (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1918pm2.43b 52 1 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  wrex 2801   class class class wbr 4481  (class class class)co 6431  Pcnp 9440   +P cpp 9442  <P cltp 9444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6728  ax-inf2 8301
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702  df-ov 6434  df-oprab 6435  df-mpt2 6436  df-om 6839  df-1st 6939  df-2nd 6940  df-wrecs 7174  df-recs 7235  df-rdg 7273  df-1o 7327  df-oadd 7331  df-omul 7332  df-er 7509  df-ni 9453  df-pli 9454  df-mi 9455  df-lti 9456  df-plpq 9489  df-mpq 9490  df-ltpq 9491  df-enq 9492  df-nq 9493  df-erq 9494  df-plq 9495  df-mq 9496  df-1nq 9497  df-rq 9498  df-ltnq 9499  df-np 9562  df-plp 9564  df-ltp 9566
This theorem is referenced by:  ltapr  9626
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