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 Description: Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by Jim Kingdon, 11-Dec-2019.)
Assertion
Ref Expression
addcomprg ((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7131 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 7137 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
31, 2sylan 278 . . . . . . . 8 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
4 prop 7131 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 7137 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
64, 5sylan 278 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
7 addcomnqg 7037 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) = (𝑧 +Q 𝑦))
87eqeq2d 2106 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
96, 8sylan2 281 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (1st𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
109anassrs 393 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (1st𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
1110rexbidva 2388 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1211ancoms 265 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
133, 12sylan2 281 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (1st𝐵))) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1413anassrs 393 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (1st𝐵)) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1514rexbidva 2388 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
16 rexcom 2545 . . . . 5 (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦))
1715, 16syl6bb 195 . . . 4 ((𝐴P𝐵P) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)))
1817rabbidv 2622 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)})
19 elprnqu 7138 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
201, 19sylan 278 . . . . . . . 8 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
21 elprnqu 7138 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
224, 21sylan 278 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
2322, 8sylan2 281 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (2nd𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2423anassrs 393 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (2nd𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2524rexbidva 2388 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2625ancoms 265 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2720, 26sylan2 281 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (2nd𝐵))) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2827anassrs 393 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (2nd𝐵)) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2928rexbidva 2388 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
30 rexcom 2545 . . . . 5 (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦))
3129, 30syl6bb 195 . . . 4 ((𝐴P𝐵P) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)))
3231rabbidv 2622 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)})
3318, 32opeq12d 3652 . 2 ((𝐴P𝐵P) → ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)}⟩ = ⟨{𝑥Q ∣ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)}, {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)}⟩)
34 plpvlu 7194 . . 3 ((𝐵P𝐴P) → (𝐵 +P 𝐴) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)}⟩)
3534ancoms 265 . 2 ((𝐴P𝐵P) → (𝐵 +P 𝐴) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)}⟩)
36 plpvlu 7194 . 2 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = ⟨{𝑥Q ∣ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)}, {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)}⟩)
3733, 35, 363eqtr4rd 2138 1 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  ∃wrex 2371  {crab 2374  ⟨cop 3469  ‘cfv 5049  (class class class)co 5690  1st c1st 5947  2nd c2nd 5948  Qcnq 6936   +Q cplq 6938  Pcnp 6947   +P cpp 6949 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431 This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-plpq 7000  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-inp 7122  df-iplp 7124 This theorem is referenced by:  prplnqu  7276  addextpr  7277  caucvgprlemcanl  7300  caucvgprprlemnkltj  7345  caucvgprprlemnbj  7349  caucvgprprlemmu  7351  caucvgprprlemloc  7359  caucvgprprlemexbt  7362  caucvgprprlemexb  7363  caucvgprprlemaddq  7364  enrer  7378  addcmpblnr  7382  mulcmpblnrlemg  7383  ltsrprg  7390  addcomsrg  7398  mulcomsrg  7400  mulasssrg  7401  distrsrg  7402  lttrsr  7405  ltposr  7406  ltsosr  7407  0lt1sr  7408  0idsr  7410  1idsr  7411  ltasrg  7413  recexgt0sr  7416  mulgt0sr  7420  aptisr  7421  mulextsr1lem  7422  archsr  7424  srpospr  7425  prsrpos  7427  prsradd  7428  prsrlt  7429  pitonnlem1p1  7480  pitoregt0  7483  recidpirqlemcalc  7491
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