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Theorem addcanpr 10203
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 10175 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
2 eleq1 2847 . . . . 5 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → ((𝐴 +P 𝐵) ∈ P ↔ (𝐴 +P 𝐶) ∈ P))
3 dmplp 10169 . . . . . 6 dom +P = (P × P)
4 0npr 10149 . . . . . 6 ¬ ∅ ∈ P
53, 4ndmovrcl 7097 . . . . 5 ((𝐴 +P 𝐶) ∈ P → (𝐴P𝐶P))
62, 5syl6bi 245 . . . 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 10202 . . . . . . . 8 (𝐴P → (𝐵<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐴 +P 𝐶)))
9 ltapr 10202 . . . . . . . 8 (𝐴P → (𝐶<P 𝐵 ↔ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵)))
108, 9orbi12d 905 . . . . . . 7 (𝐴P → ((𝐵<P 𝐶𝐶<P 𝐵) ↔ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
1110notbid 310 . . . . . 6 (𝐴P → (¬ (𝐵<P 𝐶𝐶<P 𝐵) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
1211ad2antrr 716 . . . . 5 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (¬ (𝐵<P 𝐶𝐶<P 𝐵) ↔ ¬ ((𝐴 +P 𝐵)<P (𝐴 +P 𝐶) ∨ (𝐴 +P 𝐶)<P (𝐴 +P 𝐵))))
13 ltsopr 10189 . . . . . . 7 <P Or P
14 sotrieq 5302 . . . . . . 7 ((<P Or P ∧ (𝐵P𝐶P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
1513, 14mpan 680 . . . . . 6 ((𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
1615ad2ant2l 736 . . . . 5 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (𝐵 = 𝐶 ↔ ¬ (𝐵<P 𝐶𝐶<P 𝐵)))
17 addclpr 10175 . . . . . 6 ((𝐴P𝐶P) → (𝐴 +P 𝐶) ∈ P)
18 sotrieq 5302 . . . . . . 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 680 . . . . . 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 589 . . . . 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 303 . . . 4 (((𝐴P𝐵P) ∧ (𝐴P𝐶P)) → (𝐵 = 𝐶 ↔ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)))
2221exbiri 801 . . 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 198  wa 386  wo 836   = wceq 1601  wcel 2107   class class class wbr 4886   Or wor 5273  (class class class)co 6922  Pcnp 10016   +P cpp 10018  <P cltp 10020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-omul 7848  df-er 8026  df-ni 10029  df-pli 10030  df-mi 10031  df-lti 10032  df-plpq 10065  df-mpq 10066  df-ltpq 10067  df-enq 10068  df-nq 10069  df-erq 10070  df-plq 10071  df-mq 10072  df-1nq 10073  df-rq 10074  df-ltnq 10075  df-np 10138  df-plp 10140  df-ltp 10142
This theorem is referenced by:  enrer  10220  mulcmpblnr  10228  mulgt0sr  10262
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