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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 6937 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 6943 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
31, 2sylan 277 . . . . . . . 8 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
4 prop 6937 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 6943 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
64, 5sylan 277 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
7 addcomnqg 6843 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) = (𝑧 +Q 𝑦))
87eqeq2d 2094 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
96, 8sylan2 280 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (1st𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
109anassrs 392 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (1st𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
1110rexbidva 2371 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1211ancoms 264 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
133, 12sylan2 280 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (1st𝐵))) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1413anassrs 392 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (1st𝐵)) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1514rexbidva 2371 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
16 rexcom 2524 . . . . 5 (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦))
1715, 16syl6bb 194 . . . 4 ((𝐴P𝐵P) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)))
1817rabbidv 2601 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)})
19 elprnqu 6944 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
201, 19sylan 277 . . . . . . . 8 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
21 elprnqu 6944 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
224, 21sylan 277 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
2322, 8sylan2 280 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (2nd𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2423anassrs 392 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (2nd𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2524rexbidva 2371 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2625ancoms 264 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2720, 26sylan2 280 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (2nd𝐵))) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2827anassrs 392 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (2nd𝐵)) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2928rexbidva 2371 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
30 rexcom 2524 . . . . 5 (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦))
3129, 30syl6bb 194 . . . 4 ((𝐴P𝐵P) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)))
3231rabbidv 2601 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)})
3318, 32opeq12d 3604 . 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 7000 . . 3 ((𝐵P𝐴P) → (𝐵 +P 𝐴) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)}⟩)
3534ancoms 264 . 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 7000 . 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 2126 1 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285   ∈ wcel 1434  ∃wrex 2354  {crab 2357  ⟨cop 3425  ‘cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  Qcnq 6742   +Q cplq 6744  Pcnp 6753   +P cpp 6755 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 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366 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 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-plpq 6806  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-inp 6928  df-iplp 6930 This theorem is referenced by:  prplnqu  7082  addextpr  7083  caucvgprlemcanl  7106  caucvgprprlemnkltj  7151  caucvgprprlemnbj  7155  caucvgprprlemmu  7157  caucvgprprlemloc  7165  caucvgprprlemexbt  7168  caucvgprprlemexb  7169  caucvgprprlemaddq  7170  enrer  7184  addcmpblnr  7188  mulcmpblnrlemg  7189  ltsrprg  7196  addcomsrg  7204  mulcomsrg  7206  mulasssrg  7207  distrsrg  7208  lttrsr  7211  ltposr  7212  ltsosr  7213  0lt1sr  7214  0idsr  7216  1idsr  7217  ltasrg  7219  recexgt0sr  7222  mulgt0sr  7226  aptisr  7227  mulextsr1lem  7228  archsr  7230  srpospr  7231  prsrpos  7233  prsradd  7234  prsrlt  7235  pitonnlem1p1  7286  pitoregt0  7289  recidpirqlemcalc  7297
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