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Theorem addcomprg 7398
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))

Proof of Theorem addcomprg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7295 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 7301 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
31, 2sylan 281 . . . . . . . 8 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
4 prop 7295 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 7301 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
64, 5sylan 281 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
7 addcomnqg 7201 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) = (𝑧 +Q 𝑦))
87eqeq2d 2151 . . . . . . . . . . . 12 ((𝑦Q𝑧Q) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
96, 8sylan2 284 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (1st𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
109anassrs 397 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (1st𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
1110rexbidva 2434 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1211ancoms 266 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
133, 12sylan2 284 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (1st𝐵))) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1413anassrs 397 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (1st𝐵)) → (∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
1514rexbidva 2434 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑧 +Q 𝑦)))
16 rexcom 2595 . . . . 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 2675 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑥 = (𝑧 +Q 𝑦)})
19 elprnqu 7302 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
201, 19sylan 281 . . . . . . . 8 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
21 elprnqu 7302 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
224, 21sylan 281 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
2322, 8sylan2 284 . . . . . . . . . . 11 ((𝑦Q ∧ (𝐴P𝑧 ∈ (2nd𝐴))) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2423anassrs 397 . . . . . . . . . 10 (((𝑦Q𝐴P) ∧ 𝑧 ∈ (2nd𝐴)) → (𝑥 = (𝑦 +Q 𝑧) ↔ 𝑥 = (𝑧 +Q 𝑦)))
2524rexbidva 2434 . . . . . . . . 9 ((𝑦Q𝐴P) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2625ancoms 266 . . . . . . . 8 ((𝐴P𝑦Q) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2720, 26sylan2 284 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑦 ∈ (2nd𝐵))) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2827anassrs 397 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑦 ∈ (2nd𝐵)) → (∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
2928rexbidva 2434 . . . . 5 ((𝐴P𝐵P) → (∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧) ↔ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑧 +Q 𝑦)))
30 rexcom 2595 . . . . 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 2675 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)} = {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑥 = (𝑧 +Q 𝑦)})
3318, 32opeq12d 3713 . 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 7358 . . 3 ((𝐵P𝐴P) → (𝐵 +P 𝐴) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐵)∃𝑧 ∈ (1st𝐴)𝑥 = (𝑦 +Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐵)∃𝑧 ∈ (2nd𝐴)𝑥 = (𝑦 +Q 𝑧)}⟩)
3534ancoms 266 . 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 7358 . 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 2183 1 ((𝐴P𝐵P) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wrex 2417  {crab 2420  cop 3530  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   +Q cplq 7102  Pcnp 7111   +P cpp 7113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-plpq 7164  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-inp 7286  df-iplp 7288
This theorem is referenced by:  prplnqu  7440  addextpr  7441  caucvgprlemcanl  7464  caucvgprprlemnkltj  7509  caucvgprprlemnbj  7513  caucvgprprlemmu  7515  caucvgprprlemloc  7523  caucvgprprlemexbt  7526  caucvgprprlemexb  7527  caucvgprprlemaddq  7528  enrer  7555  addcmpblnr  7559  mulcmpblnrlemg  7560  ltsrprg  7567  addcomsrg  7575  mulcomsrg  7577  mulasssrg  7578  distrsrg  7579  lttrsr  7582  ltposr  7583  ltsosr  7584  0lt1sr  7585  0idsr  7587  1idsr  7588  ltasrg  7590  recexgt0sr  7593  mulgt0sr  7598  aptisr  7599  mulextsr1lem  7600  archsr  7602  srpospr  7603  prsrpos  7605  prsradd  7606  prsrlt  7607  ltpsrprg  7623  map2psrprg  7625  pitonnlem1p1  7666  pitoregt0  7669  recidpirqlemcalc  7677
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