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Theorem ltaprg 7275
Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.)
Assertion
Ref Expression
ltaprg ((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Proof of Theorem ltaprg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ltaprlem 7274 . . 3 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
213ad2ant3 969 . 2 ((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
3 ltexpri 7269 . . . . 5 ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ∃𝑥P ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
43adantl 272 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → ∃𝑥P ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
5 simpl1 949 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴P)
6 simprl 499 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝑥P)
7 ltaddpr 7253 . . . . . . 7 ((𝐴P𝑥P) → 𝐴<P (𝐴 +P 𝑥))
85, 6, 7syl2anc 404 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴<P (𝐴 +P 𝑥))
9 addassprg 7235 . . . . . . . . . . . 12 ((𝐶P𝐴P𝑥P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
1093com12 1150 . . . . . . . . . . 11 ((𝐴P𝐶P𝑥P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
11103expa 1146 . . . . . . . . . 10 (((𝐴P𝐶P) ∧ 𝑥P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
1211adantrr 464 . . . . . . . . 9 (((𝐴P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥)))
13 simprr 500 . . . . . . . . 9 (((𝐴P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))
1412, 13eqtr3d 2129 . . . . . . . 8 (((𝐴P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
15143adantl2 1103 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵))
16 simpl3 951 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐶P)
17 addclpr 7193 . . . . . . . . 9 ((𝐴P𝑥P) → (𝐴 +P 𝑥) ∈ P)
185, 6, 17syl2anc 404 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐴 +P 𝑥) ∈ P)
19 simpl2 950 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐵P)
20 addcanprg 7272 . . . . . . . 8 ((𝐶P ∧ (𝐴 +P 𝑥) ∈ P𝐵P) → ((𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵) → (𝐴 +P 𝑥) = 𝐵))
2116, 18, 19, 20syl3anc 1181 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → ((𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵) → (𝐴 +P 𝑥) = 𝐵))
2215, 21mpd 13 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → (𝐴 +P 𝑥) = 𝐵)
238, 22breqtrd 3891 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴<P 𝐵)
2423adantlr 462 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) ∧ (𝑥P ∧ ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵))) → 𝐴<P 𝐵)
254, 24rexlimddv 2507 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → 𝐴<P 𝐵)
2625ex 114 . 2 ((𝐴P𝐵P𝐶P) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
272, 26impbid 128 1 ((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 927   = wceq 1296  wcel 1445  wrex 2371   class class class wbr 3867  (class class class)co 5690  Pcnp 6947   +P cpp 6949  <P cltp 6951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-iplp 7124  df-iltp 7126
This theorem is referenced by:  prplnqu  7276  addextpr  7277  caucvgprlemcanl  7300  caucvgprprlemnkltj  7345  caucvgprprlemnbj  7349  caucvgprprlemmu  7351  caucvgprprlemloc  7359  caucvgprprlemexbt  7362  caucvgprprlemexb  7363  caucvgprprlemaddq  7364  caucvgprprlem1  7365  caucvgprprlem2  7366  ltsrprg  7390  gt0srpr  7391  lttrsr  7405  ltsosr  7407  ltasrg  7413  prsrlt  7429
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