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Theorem addcanprg 7076
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 7074 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))
2 3ancomb 928 . . . . . . 7 ((𝐴P𝐵P𝐶P) ↔ (𝐴P𝐶P𝐵P))
3 eqcom 2085 . . . . . . 7 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) ↔ (𝐴 +P 𝐶) = (𝐴 +P 𝐵))
42, 3anbi12i 448 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ↔ ((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)))
5 addcanprleml 7074 . . . . . 6 (((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)) → (1st𝐶) ⊆ (1st𝐵))
64, 5sylbi 119 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐶) ⊆ (1st𝐵))
71, 6eqssd 3027 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) = (1st𝐶))
8 addcanprlemu 7075 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
9 addcanprlemu 7075 . . . . . 6 (((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)) → (2nd𝐶) ⊆ (2nd𝐵))
104, 9sylbi 119 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐶) ⊆ (2nd𝐵))
118, 10eqssd 3027 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) = (2nd𝐶))
127, 11jca 300 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶)))
13 preqlu 6932 . . . . 5 ((𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
14133adant1 957 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
1514adantr 270 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
1612, 15mpbird 165 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → 𝐵 = 𝐶)
1716ex 113 1 ((𝐴P𝐵P𝐶P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wss 2984  cfv 4967  (class class class)co 5589  1st c1st 5842  2nd c2nd 5843  Pcnp 6751   +P cpp 6753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4079  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-1o 6111  df-2o 6112  df-oadd 6115  df-omul 6116  df-er 6220  df-ec 6222  df-qs 6226  df-ni 6764  df-pli 6765  df-mi 6766  df-lti 6767  df-plpq 6804  df-mpq 6805  df-enq 6807  df-nqqs 6808  df-plqqs 6809  df-mqqs 6810  df-1nqqs 6811  df-rq 6812  df-ltnqqs 6813  df-enq0 6884  df-nq0 6885  df-0nq0 6886  df-plq0 6887  df-mq0 6888  df-inp 6926  df-iplp 6928
This theorem is referenced by:  lteupri  7077  ltaprg  7079  enrer  7182  mulcmpblnr  7188  mulgt0sr  7224  srpospr  7229
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