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Theorem mulcomprg 6736
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 6631 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 6637 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (1st𝐵)) → 𝑧Q)
31, 2sylan 271 . . . . . . . 8 ((𝐵P𝑧 ∈ (1st𝐵)) → 𝑧Q)
4 prop 6631 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 6637 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
64, 5sylan 271 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7 mulcomnqg 6539 . . . . . . . . . . . . 13 ((𝑧Q𝑦Q) → (𝑧 ·Q 𝑦) = (𝑦 ·Q 𝑧))
87eqeq2d 2067 . . . . . . . . . . . 12 ((𝑧Q𝑦Q) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
96, 8sylan2 274 . . . . . . . . . . 11 ((𝑧Q ∧ (𝐴P𝑦 ∈ (1st𝐴))) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
109anassrs 386 . . . . . . . . . 10 (((𝑧Q𝐴P) ∧ 𝑦 ∈ (1st𝐴)) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
1110rexbidva 2340 . . . . . . . . 9 ((𝑧Q𝐴P) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1211ancoms 259 . . . . . . . 8 ((𝐴P𝑧Q) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
133, 12sylan2 274 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑧 ∈ (1st𝐵))) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1413anassrs 386 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑧 ∈ (1st𝐵)) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1514rexbidva 2340 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
16 rexcom 2491 . . . . 5 (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧))
1715, 16syl6bb 189 . . . 4 ((𝐴P𝐵P) → (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)))
1817rabbidv 2566 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)})
19 elprnqu 6638 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
201, 19sylan 271 . . . . . . . 8 ((𝐵P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
21 elprnqu 6638 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
224, 21sylan 271 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2322, 8sylan2 274 . . . . . . . . . . 11 ((𝑧Q ∧ (𝐴P𝑦 ∈ (2nd𝐴))) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
2423anassrs 386 . . . . . . . . . 10 (((𝑧Q𝐴P) ∧ 𝑦 ∈ (2nd𝐴)) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
2524rexbidva 2340 . . . . . . . . 9 ((𝑧Q𝐴P) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2625ancoms 259 . . . . . . . 8 ((𝐴P𝑧Q) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2720, 26sylan2 274 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑧 ∈ (2nd𝐵))) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2827anassrs 386 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑧 ∈ (2nd𝐵)) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2928rexbidva 2340 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
30 rexcom 2491 . . . . 5 (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧))
3129, 30syl6bb 189 . . . 4 ((𝐴P𝐵P) → (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)))
3231rabbidv 2566 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)})
3318, 32opeq12d 3585 . 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 6695 . . 3 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = ⟨{𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)}, {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)}⟩)
3534ancoms 259 . 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 6695 . 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 2099 1 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wrex 2324  {crab 2327  cop 3406  cfv 4930  (class class class)co 5540  1st c1st 5793  2nd c2nd 5794  Qcnq 6436   ·Q cmq 6439  Pcnp 6447   ·P cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-mqqs 6506  df-inp 6622  df-imp 6625
This theorem is referenced by:  ltmprr  6798  mulcmpblnrlemg  6883  mulcomsrg  6900  mulasssrg  6901  m1m1sr  6904  recexgt0sr  6916  mulgt0sr  6920  mulextsr1lem  6922  recidpirqlemcalc  6991
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