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Theorem addcanprg 7447
Description: Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by Jim Kingdon, 24-Dec-2019.)
Assertion
Ref Expression
addcanprg ((𝐴P𝐵P𝐶P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))

Proof of Theorem addcanprg
StepHypRef Expression
1 addcanprleml 7445 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))
2 3ancomb 971 . . . . . . 7 ((𝐴P𝐵P𝐶P) ↔ (𝐴P𝐶P𝐵P))
3 eqcom 2142 . . . . . . 7 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) ↔ (𝐴 +P 𝐶) = (𝐴 +P 𝐵))
42, 3anbi12i 456 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ↔ ((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)))
5 addcanprleml 7445 . . . . . 6 (((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)) → (1st𝐶) ⊆ (1st𝐵))
64, 5sylbi 120 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐶) ⊆ (1st𝐵))
71, 6eqssd 3118 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) = (1st𝐶))
8 addcanprlemu 7446 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
9 addcanprlemu 7446 . . . . . 6 (((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)) → (2nd𝐶) ⊆ (2nd𝐵))
104, 9sylbi 120 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐶) ⊆ (2nd𝐵))
118, 10eqssd 3118 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) = (2nd𝐶))
127, 11jca 304 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶)))
13 preqlu 7303 . . . . 5 ((𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
14133adant1 1000 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
1514adantr 274 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
1612, 15mpbird 166 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → 𝐵 = 𝐶)
1716ex 114 1 ((𝐴P𝐵P𝐶P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1332  wcel 1481  wss 3075  cfv 5130  (class class class)co 5781  1st c1st 6043  2nd c2nd 6044  Pcnp 7122   +P cpp 7124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-iplp 7299
This theorem is referenced by:  lteupri  7448  ltaprg  7450  enrer  7566  mulcmpblnr  7572  mulgt0sr  7609  srpospr  7614
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