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Theorem prsradd 7013
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 6795 . . . 4 1PP
2 addclpr 6778 . . . 4 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
31, 2mpan2 416 . . 3 (𝐴P → (𝐴 +P 1P) ∈ P)
4 addclpr 6778 . . . 4 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
51, 4mpan2 416 . . 3 (𝐵P → (𝐵 +P 1P) ∈ P)
6 addsrpr 6973 . . . . 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 6819 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
1413adantl 271 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
15 addassprg 6820 . . . . . . . 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 6778 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
1817adantl 271 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) ∈ P)
1910, 11, 12, 14, 16, 11, 18caov4d 5710 . . . . . 6 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((𝐴 +P 𝐵) +P (1P +P 1P)))
20 addclpr 6778 . . . . . . 7 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
21 addclpr 6778 . . . . . . . . 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 6819 . . . . . . 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 2114 . . . . 5 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2726oveq1d 5552 . . . 4 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P))
28 addassprg 6820 . . . . 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 1170 . . . 4 ((𝐴P𝐵P) → (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
3027, 29eqtrd 2114 . . 3 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
31 addclpr 6778 . . . . 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 6778 . . . . 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 6964 . . . . . 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 2115 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 920   = wceq 1285  wcel 1434  cop 3403  (class class class)co 5537  [cec 6163  Pcnp 6532  1Pc1p 6533   +P cpp 6534   ~R cer 6537   +R cplr 6542
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 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
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 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-i1p 6708  df-iplp 6709  df-enr 6954  df-nr 6955  df-plr 6956
This theorem is referenced by:  caucvgsrlemcau  7020  caucvgsrlemgt1  7022
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