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Theorem addcanprg 6868
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 6866 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) ⊆ (1st𝐶))
2 3ancomb 928 . . . . . . 7 ((𝐴P𝐵P𝐶P) ↔ (𝐴P𝐶P𝐵P))
3 eqcom 2084 . . . . . . 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 6866 . . . . . 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 3017 . . . 4 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (1st𝐵) = (1st𝐶))
8 addcanprlemu 6867 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ (𝐴 +P 𝐵) = (𝐴 +P 𝐶)) → (2nd𝐵) ⊆ (2nd𝐶))
9 addcanprlemu 6867 . . . . . 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 3017 . . . 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 6724 . . . . 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 2974  cfv 4932  (class class class)co 5543  1st c1st 5796  2nd c2nd 5797  Pcnp 6543   +P cpp 6545
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720
This theorem is referenced by:  lteupri  6869  ltaprg  6871  enrer  6974  mulcmpblnr  6980  mulgt0sr  7016  srpospr  7021
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