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Theorem addcanprg 6657
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 6655 . . . . 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 6655 . . . . . 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 2959 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) = (1st𝐶))
8 addcanprlemu 6656 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
9 addcanprlemu 6656 . . . . . 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 2959 . . . 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 6513 . . . . 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 2914  cfv 4863  (class class class)co 5473  1st c1st 5726  2nd c2nd 5727  Pcnp 6332   +P cpp 6334
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 3868  ax-sep 3871  ax-nul 3879  ax-pow 3923  ax-pr 3940  ax-un 4141  ax-setind 4230  ax-iinf 4272
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 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-int 3612  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-tr 3851  df-eprel 4022  df-id 4026  df-po 4029  df-iso 4030  df-iord 4074  df-on 4076  df-suc 4079  df-iom 4275  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-f 4867  df-f1 4868  df-fo 4869  df-f1o 4870  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-1st 5728  df-2nd 5729  df-recs 5881  df-irdg 5918  df-1o 5962  df-2o 5963  df-oadd 5966  df-omul 5967  df-er 6065  df-ec 6067  df-qs 6071  df-ni 6345  df-pli 6346  df-mi 6347  df-lti 6348  df-plpq 6385  df-mpq 6386  df-enq 6388  df-nqqs 6389  df-plqqs 6390  df-mqqs 6391  df-1nqqs 6392  df-rq 6393  df-ltnqqs 6394  df-enq0 6465  df-nq0 6466  df-0nq0 6467  df-plq0 6468  df-mq0 6469  df-inp 6507  df-iplp 6509
This theorem is referenced by:  lteupri  6658  ltaprg  6660  enrer  6763  mulcmpblnr  6769  mulgt0sr  6805  srpospr  6810
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