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Theorem addcanprg 6714
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 6712 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))
2 3ancomb 893 . . . . . . 7 ((𝐴P𝐵P𝐶P) ↔ (𝐴P𝐶P𝐵P))
3 eqcom 2042 . . . . . . 7 ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) ↔ (𝐴 +P 𝐶) = (𝐴 +P 𝐵))
42, 3anbi12i 433 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) ↔ ((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)))
5 addcanprleml 6712 . . . . . 6 (((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)) → (1st𝐶) ⊆ (1st𝐵))
64, 5sylbi 114 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐶) ⊆ (1st𝐵))
71, 6eqssd 2962 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) = (1st𝐶))
8 addcanprlemu 6713 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
9 addcanprlemu 6713 . . . . . 6 (((𝐴P𝐶P𝐵P) ∧ (𝐴 +P 𝐶) = (𝐴 +P 𝐵)) → (2nd𝐶) ⊆ (2nd𝐵))
104, 9sylbi 114 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐶) ⊆ (2nd𝐵))
118, 10eqssd 2962 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) = (2nd𝐶))
127, 11jca 290 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶)))
13 preqlu 6570 . . . . 5 ((𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
14133adant1 922 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
1514adantr 261 . . 3 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (𝐵 = 𝐶 ↔ ((1st𝐵) = (1st𝐶) ∧ (2nd𝐵) = (2nd𝐶))))
1612, 15mpbird 156 . 2 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → 𝐵 = 𝐶)
1716ex 108 1 ((𝐴P𝐵P𝐶P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  wss 2917  cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Pcnp 6389   +P cpp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566
This theorem is referenced by:  lteupri  6715  ltaprg  6717  enrer  6820  mulcmpblnr  6826  mulgt0sr  6862  srpospr  6867
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