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Theorem prsradd 7784
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 7552 . . . 4 1PP
2 addclpr 7535 . . . 4 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
31, 2mpan2 425 . . 3 (𝐴P → (𝐴 +P 1P) ∈ P)
4 addclpr 7535 . . . 4 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
51, 4mpan2 425 . . 3 (𝐵P → (𝐵 +P 1P) ∈ P)
6 addsrpr 7743 . . . . 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 435 . . . 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 438 . . 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 289 . 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 109 . . . . . . 7 ((𝐴P𝐵P) → 𝐴P)
111a1i 9 . . . . . . 7 ((𝐴P𝐵P) → 1PP)
12 simpr 110 . . . . . . 7 ((𝐴P𝐵P) → 𝐵P)
13 addcomprg 7576 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
1413adantl 277 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
15 addassprg 7577 . . . . . . . 8 ((𝑓P𝑔PP) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
1615adantl 277 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔PP)) → ((𝑓 +P 𝑔) +P ) = (𝑓 +P (𝑔 +P )))
17 addclpr 7535 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) ∈ P)
1817adantl 277 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) ∈ P)
1910, 11, 12, 14, 16, 11, 18caov4d 6058 . . . . . 6 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((𝐴 +P 𝐵) +P (1P +P 1P)))
20 addclpr 7535 . . . . . . 7 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
21 addclpr 7535 . . . . . . . . 9 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
221, 1, 21mp2an 426 . . . . . . . 8 (1P +P 1P) ∈ P
2322a1i 9 . . . . . . 7 ((𝐴P𝐵P) → (1P +P 1P) ∈ P)
24 addcomprg 7576 . . . . . . 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 411 . . . . . 6 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) +P (1P +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2619, 25eqtrd 2210 . . . . 5 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) = ((1P +P 1P) +P (𝐴 +P 𝐵)))
2726oveq1d 5889 . . . 4 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P))
28 addassprg 7577 . . . . 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 1238 . . . 4 ((𝐴P𝐵P) → (((1P +P 1P) +P (𝐴 +P 𝐵)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
3027, 29eqtrd 2210 . . 3 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P (𝐵 +P 1P)) +P 1P) = ((1P +P 1P) +P ((𝐴 +P 𝐵) +P 1P)))
31 addclpr 7535 . . . . 5 (((𝐴 +P 1P) ∈ P ∧ (𝐵 +P 1P) ∈ P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
323, 5, 31syl2an 289 . . . 4 ((𝐴P𝐵P) → ((𝐴 +P 1P) +P (𝐵 +P 1P)) ∈ P)
33 addclpr 7535 . . . . 5 (((𝐴 +P 𝐵) ∈ P ∧ 1PP) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
3420, 11, 33syl2anc 411 . . . 4 ((𝐴P𝐵P) → ((𝐴 +P 𝐵) +P 1P) ∈ P)
35 enreceq 7734 . . . . . 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 438 . . . . 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 435 . . . 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 411 . . 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 167 . 2 ((𝐴P𝐵P) → [⟨((𝐴 +P 1P) +P (𝐵 +P 1P)), (1P +P 1P)⟩] ~R = [⟨((𝐴 +P 𝐵) +P 1P), 1P⟩] ~R )
409, 39eqtr2d 2211 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 104  wb 105  w3a 978   = wceq 1353  wcel 2148  cop 3595  (class class class)co 5874  [cec 6532  Pcnp 7289  1Pc1p 7290   +P cpp 7291   ~R cer 7294   +R cplr 7299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-iplp 7466  df-enr 7724  df-nr 7725  df-plr 7726
This theorem is referenced by:  caucvgsrlemcau  7791  caucvgsrlemgt1  7793
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