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Theorem prsradd 7562
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 7330 . . . 4 1PP
2 addclpr 7313 . . . 4 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
31, 2mpan2 421 . . 3 (𝐴P → (𝐴 +P 1P) ∈ P)
4 addclpr 7313 . . . 4 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
51, 4mpan2 421 . . 3 (𝐵P → (𝐵 +P 1P) ∈ P)
6 addsrpr 7521 . . . . 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 431 . . . 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 434 . . 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 287 . 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 108 . . . . . . 7 ((𝐴P𝐵P) → 𝐴P)
111a1i 9 . . . . . . 7 ((𝐴P𝐵P) → 1PP)
12 simpr 109 . . . . . . 7 ((𝐴P𝐵P) → 𝐵P)
13 addcomprg 7354 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
1413adantl 275 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
15 addassprg 7355 . . . . . . . 8 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
1615adantl 275 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
17 addclpr 7313 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
1817adantl 275 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) ∈ P)
1910, 11, 12, 14, 16, 11, 18caov4d 5923 . . . . . 6 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((𝐴 +P 𝐵) +P (1P +P 1P)))
20 addclpr 7313 . . . . . . 7 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
21 addclpr 7313 . . . . . . . . 9 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
221, 1, 21mp2an 422 . . . . . . . 8 (1P +P 1P) ∈ P
2322a1i 9 . . . . . . 7 ((𝐴P𝐵P) → (1P +P 1P) ∈ P)
24 addcomprg 7354 . . . . . . 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 408 . . . . . 6 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) +P (1P +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2619, 25eqtrd 2150 . . . . 5 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2726oveq1d 5757 . . . 4 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P))
28 addassprg 7355 . . . . 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 1201 . . . 4 ((𝐴P𝐵P) → (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
3027, 29eqtrd 2150 . . 3 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
31 addclpr 7313 . . . . 5 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
323, 5, 31syl2an 287 . . . 4 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
33 addclpr 7313 . . . . 5 (((𝐴 +P 𝐵) ∈ P ∧ 1PP) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
3420, 11, 33syl2anc 408 . . . 4 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
35 enreceq 7512 . . . . . 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 434 . . . . 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 431 . . . 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 408 . . 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 166 . 2 ((𝐴P𝐵P) → [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R )
409, 39eqtr2d 2151 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 103  wb 104  w3a 947   = wceq 1316  wcel 1465  cop 3500  (class class class)co 5742  [cec 6395  Pcnp 7067  1Pc1p 7068   +P cpp 7069   ~R cer 7072   +R cplr 7077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-iplp 7244  df-enr 7502  df-nr 7503  df-plr 7504
This theorem is referenced by:  caucvgsrlemcau  7569  caucvgsrlemgt1  7571
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