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Theorem addcanprg 6590
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 ((A P B P 𝐶 P) → ((A +P B) = (A +P 𝐶) → B = 𝐶))

Proof of Theorem addcanprg
StepHypRef Expression
1 addcanprleml 6588 . . . . 5 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) ⊆ (1st𝐶))
2 3ancomb 892 . . . . . . 7 ((A P B P 𝐶 P) ↔ (A P 𝐶 P B P))
3 eqcom 2039 . . . . . . 7 ((A +P B) = (A +P 𝐶) ↔ (A +P 𝐶) = (A +P B))
42, 3anbi12i 433 . . . . . 6 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) ↔ ((A P 𝐶 P B P) (A +P 𝐶) = (A +P B)))
5 addcanprleml 6588 . . . . . 6 (((A P 𝐶 P B P) (A +P 𝐶) = (A +P B)) → (1st𝐶) ⊆ (1stB))
64, 5sylbi 114 . . . . 5 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1st𝐶) ⊆ (1stB))
71, 6eqssd 2956 . . . 4 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) = (1st𝐶))
8 addcanprlemu 6589 . . . . 5 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2ndB) ⊆ (2nd𝐶))
9 addcanprlemu 6589 . . . . . 6 (((A P 𝐶 P B P) (A +P 𝐶) = (A +P B)) → (2nd𝐶) ⊆ (2ndB))
104, 9sylbi 114 . . . . 5 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2nd𝐶) ⊆ (2ndB))
118, 10eqssd 2956 . . . 4 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2ndB) = (2nd𝐶))
127, 11jca 290 . . 3 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → ((1stB) = (1st𝐶) (2ndB) = (2nd𝐶)))
13 preqlu 6455 . . . . 5 ((B P 𝐶 P) → (B = 𝐶 ↔ ((1stB) = (1st𝐶) (2ndB) = (2nd𝐶))))
14133adant1 921 . . . 4 ((A P B P 𝐶 P) → (B = 𝐶 ↔ ((1stB) = (1st𝐶) (2ndB) = (2nd𝐶))))
1514adantr 261 . . 3 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (B = 𝐶 ↔ ((1stB) = (1st𝐶) (2ndB) = (2nd𝐶))))
1612, 15mpbird 156 . 2 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → B = 𝐶)
1716ex 108 1 ((A P B P 𝐶 P) → ((A +P B) = (A +P 𝐶) → B = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wss 2911  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Pcnp 6275   +P cpp 6277
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451
This theorem is referenced by:  ltaprg  6592  enrer  6663  mulcmpblnr  6669  mulgt0sr  6704
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