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Theorem addcanprg 6604
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 6602 . . . . 5 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) ⊆ (1st𝐶))
2 3ancomb 893 . . . . . . 7 ((A P B P 𝐶 P) ↔ (A P 𝐶 P B P))
3 eqcom 2042 . . . . . . 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 6602 . . . . . 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 2959 . . . 4 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (1stB) = (1st𝐶))
8 addcanprlemu 6603 . . . . 5 (((A P B P 𝐶 P) (A +P B) = (A +P 𝐶)) → (2ndB) ⊆ (2nd𝐶))
9 addcanprlemu 6603 . . . . . 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 2959 . . . 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 6460 . . . . 5 ((B P 𝐶 P) → (B = 𝐶 ↔ ((1stB) = (1st𝐶) (2ndB) = (2nd𝐶))))
14133adant1 922 . . . 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 885   = wceq 1243   wcel 1393  wss 2914  cfv 4848  (class class class)co 5458  1st c1st 5710  2nd c2nd 5711  Pcnp 6279   +P cpp 6281
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 3866  ax-sep 3869  ax-nul 3877  ax-pow 3921  ax-pr 3938  ax-un 4139  ax-setind 4223  ax-iinf 4257
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 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-int 3610  df-iun 3653  df-br 3759  df-opab 3813  df-mpt 3814  df-tr 3849  df-eprel 4020  df-id 4024  df-po 4027  df-iso 4028  df-iord 4072  df-on 4074  df-suc 4077  df-iom 4260  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-rn 4302  df-res 4303  df-ima 4304  df-iota 4813  df-fun 4850  df-fn 4851  df-f 4852  df-f1 4853  df-fo 4854  df-f1o 4855  df-fv 4856  df-ov 5461  df-oprab 5462  df-mpt2 5463  df-1st 5712  df-2nd 5713  df-recs 5865  df-irdg 5901  df-1o 5944  df-2o 5945  df-oadd 5948  df-omul 5949  df-er 6046  df-ec 6048  df-qs 6052  df-ni 6292  df-pli 6293  df-mi 6294  df-lti 6295  df-plpq 6332  df-mpq 6333  df-enq 6335  df-nqqs 6336  df-plqqs 6337  df-mqqs 6338  df-1nqqs 6339  df-rq 6340  df-ltnqqs 6341  df-enq0 6412  df-nq0 6413  df-0nq0 6414  df-plq0 6415  df-mq0 6416  df-inp 6454  df-iplp 6456
This theorem is referenced by:  lteupri  6605  ltaprg  6607  enrer  6710  mulcmpblnr  6716  mulgt0sr  6752  srpospr  6757
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