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Theorem addcanpr 9556
Description: Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
addcanpr ((𝐴P𝐵P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))

Proof of Theorem addcanpr
StepHypRef Expression
1 addclpr 9528 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2 eleq1 2571 . . . . 5 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P ↔ (𝐴 +P 𝐶) ∈ P))
3 dmplp 9522 . . . . . 6 dom +P = (P × P)
4 0npr 9502 . . . . . 6 ¬ ∅ ∈ P
53, 4ndmovrcl 6530 . . . . 5 ((𝐴 +P 𝐶) ∈ P → (𝐴P𝐶P))
62, 5syl6bi 238 . . . 4 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P → (𝐴P𝐶P)))
71, 6syl5com 31 . . 3 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → (𝐴P𝐶P)))
8 ltapr 9555 . . . . . . . 8 (𝐴P → (𝐵<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐴 +P 𝐶)))
9 ltapr 9555 . . . . . . . 8 (𝐴P → (𝐶<P 𝐵 ↔ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵)))
108, 9orbi12d 733 . . . . . . 7 (𝐴P → ((𝐵<P 𝐶𝐶<P 𝐵) ↔ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
1110notbid 303 . . . . . 6 (𝐴P → (¬ (𝐵<P 𝐶𝐶<P 𝐵) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
1211ad2antrr 749 . . . . 5 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (¬ (𝐵<P 𝐶𝐶<P 𝐵) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
13 ltsopr 9542 . . . . . . 7 <P Or P
14 sotrieq 4828 . . . . . . 7 ((<P Or P ∧ (𝐵P𝐶P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
1513, 14mpan 693 . . . . . 6 ((𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
1615ad2ant2l 769 . . . . 5 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
17 addclpr 9528 . . . . . 6 ((𝐴P𝐶P) → (𝐴 +P 𝐶) ∈ P)
18 sotrieq 4828 . . . . . . 7 ((<P Or P ∧ ((𝐴 +P 𝐵) ∈ P ∧ (𝐴 +P 𝐶) ∈ P)) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
1913, 18mpan 693 . . . . . 6 (((𝐴 +P 𝐵) ∈ P ∧ (𝐴 +P 𝐶) ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
201, 17, 19syl2an 487 . . . . 5 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
2112, 16, 203bitr4d 295 . . . 4 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (𝐵 = 𝐶 ↔ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)))
2221exbiri 641 . . 3 ((𝐴P𝐵P) → ((𝐴P𝐶P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
237, 22syld 45 . 2 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
2423pm2.43d 50 1 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 191  wo 377  wa 378   = wceq 1468  wcel 1937   class class class wbr 4434   Or wor 4800  (class class class)co 6363  Pcnp 9369   +P cpp 9371  <P cltp 9373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pow 4619  ax-pr 4680  ax-un 6659  ax-inf2 8231
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3068  df-sbc 3292  df-csb 3386  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-int 4265  df-iun 4309  df-br 4435  df-opab 4494  df-mpt 4495  df-tr 4531  df-eprel 4791  df-id 4795  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-pred 5431  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-iota 5597  df-fun 5635  df-fn 5636  df-f 5637  df-f1 5638  df-fo 5639  df-f1o 5640  df-fv 5641  df-ov 6366  df-oprab 6367  df-mpt2 6368  df-om 6770  df-1st 6870  df-2nd 6871  df-wrecs 7105  df-recs 7167  df-rdg 7205  df-1o 7259  df-oadd 7263  df-omul 7264  df-er 7440  df-ni 9382  df-pli 9383  df-mi 9384  df-lti 9385  df-plpq 9418  df-mpq 9419  df-ltpq 9420  df-enq 9421  df-nq 9422  df-erq 9423  df-plq 9424  df-mq 9425  df-1nq 9426  df-rq 9427  df-ltnq 9428  df-np 9491  df-plp 9493  df-ltp 9495
This theorem is referenced by:  enrer  9574  mulcmpblnr  9580  mulgt0sr  9614
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