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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 6573 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 6579 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
31, 2sylan 267 . . . . . . . 8 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
4 prop 6573 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 6579 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
64, 5sylan 267 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
7 addcomnqg 6479 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) = (𝑧 +Q 𝑦))
87eqeq2d 2051 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
96, 8sylan2 270 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (1st𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
109anassrs 380 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (1st𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
1110rexbidva 2323 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1211ancoms 255 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
133, 12sylan2 270 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (1st𝐵))) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1413anassrs 380 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (1st𝐵)) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1514rexbidva 2323 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
16 rexcom 2474 . . . . 5 (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦))
1715, 16syl6bb 185 . . . 4 ((𝐴P𝐵P) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)))
1817rabbidv 2549 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)})
19 elprnqu 6580 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
201, 19sylan 267 . . . . . . . 8 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
21 elprnqu 6580 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
224, 21sylan 267 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
2322, 8sylan2 270 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (2nd𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2423anassrs 380 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (2nd𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2524rexbidva 2323 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2625ancoms 255 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2720, 26sylan2 270 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (2nd𝐵))) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2827anassrs 380 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (2nd𝐵)) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2928rexbidva 2323 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
30 rexcom 2474 . . . . 5 (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦))
3129, 30syl6bb 185 . . . 4 ((𝐴P𝐵P) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)))
3231rabbidv 2549 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)})
3318, 32opeq12d 3557 . 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 6636 . . 3 ((𝐵P𝐴P) → (𝐵 +P 𝐴) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)}⟩)
3534ancoms 255 . 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 6636 . 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 2083 1 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  {crab 2310  ⟨cop 3378  ‘cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Qcnq 6378   +Q cplq 6380  Pcnp 6389   +P cpp 6391 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-plpq 6442  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-inp 6564  df-iplp 6566 This theorem is referenced by:  prplnqu  6718  addextpr  6719  caucvgprlemcanl  6742  caucvgprprlemnkltj  6787  caucvgprprlemnbj  6791  caucvgprprlemmu  6793  caucvgprprlemloc  6801  caucvgprprlemexbt  6804  caucvgprprlemexb  6805  caucvgprprlemaddq  6806  enrer  6820  addcmpblnr  6824  mulcmpblnrlemg  6825  ltsrprg  6832  addcomsrg  6840  mulcomsrg  6842  mulasssrg  6843  distrsrg  6844  lttrsr  6847  ltposr  6848  ltsosr  6849  0lt1sr  6850  0idsr  6852  1idsr  6853  ltasrg  6855  recexgt0sr  6858  mulgt0sr  6862  aptisr  6863  mulextsr1lem  6864  archsr  6866  srpospr  6867  prsrpos  6869  prsradd  6870  prsrlt  6871  pitonnlem1p1  6922  pitoregt0  6925  recidpirqlemcalc  6933
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