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Theorem mulcomprg 7609
Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.)
Assertion
Ref Expression
mulcomprg ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))

Proof of Theorem mulcomprg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7504 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 7510 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (1st𝐵)) → 𝑧Q)
31, 2sylan 283 . . . . . . . 8 ((𝐵P𝑧 ∈ (1st𝐵)) → 𝑧Q)
4 prop 7504 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 7510 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
64, 5sylan 283 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7 mulcomnqg 7412 . . . . . . . . . . . . 13 ((𝑧Q𝑦Q) → (𝑧 ·Q 𝑦) = (𝑦 ·Q 𝑧))
87eqeq2d 2201 . . . . . . . . . . . 12 ((𝑧Q𝑦Q) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
96, 8sylan2 286 . . . . . . . . . . 11 ((𝑧Q ∧ (𝐴P𝑦 ∈ (1st𝐴))) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
109anassrs 400 . . . . . . . . . 10 (((𝑧Q𝐴P) ∧ 𝑦 ∈ (1st𝐴)) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
1110rexbidva 2487 . . . . . . . . 9 ((𝑧Q𝐴P) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1211ancoms 268 . . . . . . . 8 ((𝐴P𝑧Q) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
133, 12sylan2 286 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑧 ∈ (1st𝐵))) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1413anassrs 400 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑧 ∈ (1st𝐵)) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1514rexbidva 2487 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
16 rexcom 2654 . . . . 5 (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧))
1715, 16bitrdi 196 . . . 4 ((𝐴P𝐵P) → (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)))
1817rabbidv 2741 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)})
19 elprnqu 7511 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
201, 19sylan 283 . . . . . . . 8 ((𝐵P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
21 elprnqu 7511 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
224, 21sylan 283 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2322, 8sylan2 286 . . . . . . . . . . 11 ((𝑧Q ∧ (𝐴P𝑦 ∈ (2nd𝐴))) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
2423anassrs 400 . . . . . . . . . 10 (((𝑧Q𝐴P) ∧ 𝑦 ∈ (2nd𝐴)) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
2524rexbidva 2487 . . . . . . . . 9 ((𝑧Q𝐴P) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2625ancoms 268 . . . . . . . 8 ((𝐴P𝑧Q) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2720, 26sylan2 286 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑧 ∈ (2nd𝐵))) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2827anassrs 400 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑧 ∈ (2nd𝐵)) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2928rexbidva 2487 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
30 rexcom 2654 . . . . 5 (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧))
3129, 30bitrdi 196 . . . 4 ((𝐴P𝐵P) → (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)))
3231rabbidv 2741 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)})
3318, 32opeq12d 3801 . 2 ((𝐴P𝐵P) → ⟨{𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)}, {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)}⟩ = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)}⟩)
34 mpvlu 7568 . . 3 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = ⟨{𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)}, {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)}⟩)
3534ancoms 268 . 2 ((𝐴P𝐵P) → (𝐵 ·P 𝐴) = ⟨{𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)}, {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)}⟩)
36 mpvlu 7568 . 2 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = ⟨{𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)}, {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)}⟩)
3733, 35, 363eqtr4rd 2233 1 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wrex 2469  {crab 2472  cop 3610  cfv 5235  (class class class)co 5896  1st c1st 6163  2nd c2nd 6164  Qcnq 7309   ·Q cmq 7312  Pcnp 7320   ·P cmp 7323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-mi 7335  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-mqqs 7379  df-inp 7495  df-imp 7498
This theorem is referenced by:  ltmprr  7671  mulcmpblnrlemg  7769  mulcomsrg  7786  mulasssrg  7787  m1m1sr  7790  recexgt0sr  7802  mulgt0sr  7807  mulextsr1lem  7809  recidpirqlemcalc  7886
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