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Theorem prsradd 7278
Description: Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
Assertion
Ref Expression
prsradd ((𝐴P𝐵P) → [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R = ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ))

Proof of Theorem prsradd
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pr 7060 . . . 4 1PP
2 addclpr 7043 . . . 4 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
31, 2mpan2 416 . . 3 (𝐴P → (𝐴 +P 1P) ∈ P)
4 addclpr 7043 . . . 4 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
51, 4mpan2 416 . . 3 (𝐵P → (𝐵 +P 1P) ∈ P)
6 addsrpr 7238 . . . . 5 ((((𝐴 +P 1P) ∈ P ∧ 1PP) ∧ ((𝐵 +P 1P) ∈ P ∧ 1PP)) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
71, 6mpanl2 426 . . . 4 (((𝐴 +P 1P) ∈ P ∧ ((𝐵 +P 1P) ∈ P ∧ 1PP)) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
81, 7mpanr2 429 . . 3 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
93, 5, 8syl2an 283 . 2 ((𝐴P𝐵P) → ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ) = [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R )
10 simpl 107 . . . . . . 7 ((𝐴P𝐵P) → 𝐴P)
111a1i 9 . . . . . . 7 ((𝐴P𝐵P) → 1PP)
12 simpr 108 . . . . . . 7 ((𝐴P𝐵P) → 𝐵P)
13 addcomprg 7084 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
1413adantl 271 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
15 addassprg 7085 . . . . . . . 8 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
1615adantl 271 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
17 addclpr 7043 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
1817adantl 271 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) ∈ P)
1910, 11, 12, 14, 16, 11, 18caov4d 5788 . . . . . 6 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((𝐴 +P 𝐵) +P (1P +P 1P)))
20 addclpr 7043 . . . . . . 7 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
21 addclpr 7043 . . . . . . . . 9 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
221, 1, 21mp2an 417 . . . . . . . 8 (1P +P 1P) ∈ P
2322a1i 9 . . . . . . 7 ((𝐴P𝐵P) → (1P +P 1P) ∈ P)
24 addcomprg 7084 . . . . . . 7 (((𝐴 +P 𝐵) ∈ P ∧ (1P +P 1P) ∈ P) → ((𝐴 +P 𝐵) +P (1P +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2520, 23, 24syl2anc 403 . . . . . 6 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) +P (1P +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2619, 25eqtrd 2117 . . . . 5 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2726oveq1d 5630 . . . 4 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P))
28 addassprg 7085 . . . . 5 (((1P +P 1P) ∈ P ∧ (𝐴 +P 𝐵) ∈ P ∧ 1PP) → (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
2923, 20, 11, 28syl3anc 1172 . . . 4 ((𝐴P𝐵P) → (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
3027, 29eqtrd 2117 . . 3 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
31 addclpr 7043 . . . . 5 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
323, 5, 31syl2an 283 . . . 4 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
33 addclpr 7043 . . . . 5 (((𝐴 +P 𝐵) ∈ P ∧ 1PP) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
3420, 11, 33syl2anc 403 . . . 4 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
35 enreceq 7229 . . . . . 6 (((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ (1P +P 1P) ∈ P) ∧ (((𝐴 +P 𝐵) +P 1P) ∈ P ∧ 1PP)) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
361, 35mpanr2 429 . . . . 5 (((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ (1P +P 1P) ∈ P) ∧ ((𝐴 +P 𝐵) +P 1P) ∈ P) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
3722, 36mpanl2 426 . . . 4 ((((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P ∧ ((𝐴 +P 𝐵) +P 1P) ∈ P) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
3832, 34, 37syl2anc 403 . . 3 ((𝐴P𝐵P) → ([⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R ↔ (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P))))
3930, 38mpbird 165 . 2 ((𝐴P𝐵P) → [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R )
409, 39eqtr2d 2118 1 ((𝐴P𝐵P) → [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R = ([⟨(𝐴 +P 1P), 1P⟩] ~R +R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922   = wceq 1287  wcel 1436  cop 3434  (class class class)co 5615  [cec 6244  Pcnp 6797  1Pc1p 6798   +P cpp 6799   ~R cer 6802   +R cplr 6807
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-po 4099  df-iso 4100  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-2o 6138  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-enq0 6930  df-nq0 6931  df-0nq0 6932  df-plq0 6933  df-mq0 6934  df-inp 6972  df-i1p 6973  df-iplp 6974  df-enr 7219  df-nr 7220  df-plr 7221
This theorem is referenced by:  caucvgsrlemcau  7285  caucvgsrlemgt1  7287
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