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Theorem prsrlem1 6630
Description: Decomposing signed reals into positive reals. Lemma for addsrpr 6633 and mulsrpr 6634. (Contributed by Jim Kingdon, 30-Dec-2019.)
Assertion
Ref Expression
prsrlem1 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ((((w P v P) (𝑠 P f P)) ((u P 𝑡 P) (g P P))) ((w +P f) = (v +P 𝑠) (u +P ) = (𝑡 +P g))))
Distinct variable group:   f,g,,𝑠,𝑡,u,v,w
Allowed substitution hints:   A(w,v,u,𝑡,f,g,,𝑠)   B(w,v,u,𝑡,f,g,,𝑠)

Proof of Theorem prsrlem1
Dummy variables 𝑎 𝑏 𝑐 𝑑 x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrer 6623 . . . . . 6 ~R Er (P × P)
2 erdm 6052 . . . . . 6 ( ~R Er (P × P) → dom ~R = (P × P))
31, 2ax-mp 7 . . . . 5 dom ~R = (P × P)
4 simprll 489 . . . . . 6 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → A = [⟨w, v⟩] ~R )
5 simpll 481 . . . . . 6 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → A ((P × P) / ~R ))
64, 5eqeltrrd 2112 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨w, v⟩] ~R ((P × P) / ~R ))
7 ecelqsdm 6112 . . . . 5 ((dom ~R = (P × P) [⟨w, v⟩] ~R ((P × P) / ~R )) → ⟨w, v (P × P))
83, 6, 7sylancr 393 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ⟨w, v (P × P))
9 opelxp 4317 . . . 4 (⟨w, v (P × P) ↔ (w P v P))
108, 9sylib 127 . . 3 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (w P v P))
11 simprrl 491 . . . . . 6 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → A = [⟨𝑠, f⟩] ~R )
1211, 5eqeltrrd 2112 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨𝑠, f⟩] ~R ((P × P) / ~R ))
13 ecelqsdm 6112 . . . . 5 ((dom ~R = (P × P) [⟨𝑠, f⟩] ~R ((P × P) / ~R )) → ⟨𝑠, f (P × P))
143, 12, 13sylancr 393 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ⟨𝑠, f (P × P))
15 opelxp 4317 . . . 4 (⟨𝑠, f (P × P) ↔ (𝑠 P f P))
1614, 15sylib 127 . . 3 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (𝑠 P f P))
1710, 16jca 290 . 2 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ((w P v P) (𝑠 P f P)))
18 simprlr 490 . . . . . 6 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → B = [⟨u, 𝑡⟩] ~R )
19 simplr 482 . . . . . 6 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → B ((P × P) / ~R ))
2018, 19eqeltrrd 2112 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨u, 𝑡⟩] ~R ((P × P) / ~R ))
21 ecelqsdm 6112 . . . . 5 ((dom ~R = (P × P) [⟨u, 𝑡⟩] ~R ((P × P) / ~R )) → ⟨u, 𝑡 (P × P))
223, 20, 21sylancr 393 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ⟨u, 𝑡 (P × P))
23 opelxp 4317 . . . 4 (⟨u, 𝑡 (P × P) ↔ (u P 𝑡 P))
2422, 23sylib 127 . . 3 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (u P 𝑡 P))
25 simprrr 492 . . . . . 6 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → B = [⟨g, ⟩] ~R )
2625, 19eqeltrrd 2112 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨g, ⟩] ~R ((P × P) / ~R ))
27 ecelqsdm 6112 . . . . 5 ((dom ~R = (P × P) [⟨g, ⟩] ~R ((P × P) / ~R )) → ⟨g, (P × P))
283, 26, 27sylancr 393 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ⟨g, (P × P))
29 opelxp 4317 . . . 4 (⟨g, (P × P) ↔ (g P P))
3028, 29sylib 127 . . 3 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (g P P))
3124, 30jca 290 . 2 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ((u P 𝑡 P) (g P P)))
324, 11eqtr3d 2071 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨w, v⟩] ~R = [⟨𝑠, f⟩] ~R )
331a1i 9 . . . . . 6 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ~R Er (P × P))
3433, 8erth 6086 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (⟨w, v⟩ ~R𝑠, f⟩ ↔ [⟨w, v⟩] ~R = [⟨𝑠, f⟩] ~R ))
3532, 34mpbird 156 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ⟨w, v⟩ ~R𝑠, f⟩)
36 df-enr 6614 . . . . . 6 ~R = {⟨x, y⟩ ∣ ((x (P × P) y (P × P)) 𝑎𝑏𝑐𝑑((x = ⟨𝑎, 𝑏 y = ⟨𝑐, 𝑑⟩) (𝑎 +P 𝑑) = (𝑏 +P 𝑐)))}
3736ecopoveq 6137 . . . . 5 (((w P v P) (𝑠 P f P)) → (⟨w, v⟩ ~R𝑠, f⟩ ↔ (w +P f) = (v +P 𝑠)))
3810, 16, 37syl2anc 391 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (⟨w, v⟩ ~R𝑠, f⟩ ↔ (w +P f) = (v +P 𝑠)))
3935, 38mpbid 135 . . 3 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (w +P f) = (v +P 𝑠))
4018, 25eqtr3d 2071 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → [⟨u, 𝑡⟩] ~R = [⟨g, ⟩] ~R )
4133, 22erth 6086 . . . . 5 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (⟨u, 𝑡⟩ ~Rg, ⟩ ↔ [⟨u, 𝑡⟩] ~R = [⟨g, ⟩] ~R ))
4240, 41mpbird 156 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ⟨u, 𝑡⟩ ~Rg, ⟩)
4336ecopoveq 6137 . . . . 5 (((u P 𝑡 P) (g P P)) → (⟨u, 𝑡⟩ ~Rg, ⟩ ↔ (u +P ) = (𝑡 +P g)))
4424, 30, 43syl2anc 391 . . . 4 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (⟨u, 𝑡⟩ ~Rg, ⟩ ↔ (u +P ) = (𝑡 +P g)))
4542, 44mpbid 135 . . 3 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → (u +P ) = (𝑡 +P g))
4639, 45jca 290 . 2 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ((w +P f) = (v +P 𝑠) (u +P ) = (𝑡 +P g)))
4717, 31, 46jca31 292 1 (((A ((P × P) / ~R ) B ((P × P) / ~R )) ((A = [⟨w, v⟩] ~R B = [⟨u, 𝑡⟩] ~R ) (A = [⟨𝑠, f⟩] ~R B = [⟨g, ⟩] ~R ))) → ((((w P v P) (𝑠 P f P)) ((u P 𝑡 P) (g P P))) ((w +P f) = (v +P 𝑠) (u +P ) = (𝑡 +P g))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755   × cxp 4286  dom cdm 4288  (class class class)co 5455   Er wer 6039  [cec 6040   / cqs 6041  Pcnp 6275   +P cpp 6277   ~R cer 6280
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450  df-enr 6614
This theorem is referenced by:  addsrmo  6631  mulsrmo  6632
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