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Theorem addcanpr 10446
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 10418 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2 eleq1 2898 . . . . 5 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P ↔ (𝐴 +P 𝐶) ∈ P))
3 dmplp 10412 . . . . . 6 dom +P = (P × P)
4 0npr 10392 . . . . . 6 ¬ ∅ ∈ P
53, 4ndmovrcl 7312 . . . . 5 ((𝐴 +P 𝐶) ∈ P → (𝐴P𝐶P))
62, 5syl6bi 255 . . . 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 10445 . . . . . . . 8 (𝐴P → (𝐵<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐴 +P 𝐶)))
9 ltapr 10445 . . . . . . . 8 (𝐴P → (𝐶<P 𝐵 ↔ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵)))
108, 9orbi12d 915 . . . . . . 7 (𝐴P → ((𝐵<P 𝐶𝐶<P 𝐵) ↔ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
1110notbid 320 . . . . . 6 (𝐴P → (¬ (𝐵<P 𝐶𝐶<P 𝐵) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
1211ad2antrr 724 . . . . 5 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (¬ (𝐵<P 𝐶𝐶<P 𝐵) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
13 ltsopr 10432 . . . . . . 7 <P Or P
14 sotrieq 5478 . . . . . . 7 ((<P Or P ∧ (𝐵P𝐶P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
1513, 14mpan 688 . . . . . 6 ((𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
1615ad2ant2l 744 . . . . 5 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
17 addclpr 10418 . . . . . 6 ((𝐴P𝐶P) → (𝐴 +P 𝐶) ∈ P)
18 sotrieq 5478 . . . . . . 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 688 . . . . . 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 597 . . . . 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 313 . . . 4 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (𝐵 = 𝐶 ↔ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)))
2221exbiri 809 . . 3 ((𝐴P𝐵P) → ((𝐴P𝐶P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
237, 22syld 47 . 2 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)))
2423pm2.43d 53 1 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1537  wcel 2114   class class class wbr 5042   Or wor 5449  (class class class)co 7133  Pcnp 10259   +P cpp 10261  <P cltp 10263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-omul 8085  df-er 8267  df-ni 10272  df-pli 10273  df-mi 10274  df-lti 10275  df-plpq 10308  df-mpq 10309  df-ltpq 10310  df-enq 10311  df-nq 10312  df-erq 10313  df-plq 10314  df-mq 10315  df-1nq 10316  df-rq 10317  df-ltnq 10318  df-np 10381  df-plp 10383  df-ltp 10385
This theorem is referenced by:  enrer  10463  mulcmpblnr  10471  mulgt0sr  10505
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