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Theorem mulcomprg 6635
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 6530 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 6536 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (1st𝐵)) → 𝑧Q)
31, 2sylan 267 . . . . . . . 8 ((𝐵P𝑧 ∈ (1st𝐵)) → 𝑧Q)
4 prop 6530 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 6536 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
64, 5sylan 267 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7 mulcomnqg 6438 . . . . . . . . . . . . 13 ((𝑧Q𝑦Q) → (𝑧 ·Q 𝑦) = (𝑦 ·Q 𝑧))
87eqeq2d 2051 . . . . . . . . . . . 12 ((𝑧Q𝑦Q) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
96, 8sylan2 270 . . . . . . . . . . 11 ((𝑧Q ∧ (𝐴P𝑦 ∈ (1st𝐴))) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
109anassrs 380 . . . . . . . . . 10 (((𝑧Q𝐴P) ∧ 𝑦 ∈ (1st𝐴)) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
1110rexbidva 2320 . . . . . . . . 9 ((𝑧Q𝐴P) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1211ancoms 255 . . . . . . . 8 ((𝐴P𝑧Q) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
133, 12sylan2 270 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑧 ∈ (1st𝐵))) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1413anassrs 380 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑧 ∈ (1st𝐵)) → (∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
1514rexbidva 2320 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
16 rexcom 2471 . . . . 5 (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧))
1715, 16syl6bb 185 . . . 4 ((𝐴P𝐵P) → (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)))
1817rabbidv 2546 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)})
19 elprnqu 6537 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
201, 19sylan 267 . . . . . . . 8 ((𝐵P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
21 elprnqu 6537 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
224, 21sylan 267 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (2nd𝐴)) → 𝑦Q)
2322, 8sylan2 270 . . . . . . . . . . 11 ((𝑧Q ∧ (𝐴P𝑦 ∈ (2nd𝐴))) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
2423anassrs 380 . . . . . . . . . 10 (((𝑧Q𝐴P) ∧ 𝑦 ∈ (2nd𝐴)) → (𝑥 = (𝑧 ·Q 𝑦) ↔ 𝑥 = (𝑦 ·Q 𝑧)))
2524rexbidva 2320 . . . . . . . . 9 ((𝑧Q𝐴P) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2625ancoms 255 . . . . . . . 8 ((𝐴P𝑧Q) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2720, 26sylan2 270 . . . . . . 7 ((𝐴P ∧ (𝐵P𝑧 ∈ (2nd𝐵))) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2827anassrs 380 . . . . . 6 (((𝐴P𝐵P) ∧ 𝑧 ∈ (2nd𝐵)) → (∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
2928rexbidva 2320 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
30 rexcom 2471 . . . . 5 (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧))
3129, 30syl6bb 185 . . . 4 ((𝐴P𝐵P) → (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)))
3231rabbidv 2546 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)})
3318, 32opeq12d 3554 . 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 6594 . . 3 ((𝐵P𝐴P) → (𝐵 ·P 𝐴) = ⟨{𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)}, {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)}⟩)
3534ancoms 255 . 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 6594 . 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 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 2304  {crab 2307  cop 3375  cfv 4865  (class class class)co 5475  1st c1st 5728  2nd c2nd 5729  Qcnq 6335   ·Q cmq 6338  Pcnp 6346   ·P cmp 6349
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 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
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 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-id 4027  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-mi 6361  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-mqqs 6405  df-inp 6521  df-imp 6524
This theorem is referenced by:  ltmprr  6697  mulcmpblnrlemg  6782  mulcomsrg  6799  mulasssrg  6800  m1m1sr  6803  recexgt0sr  6815  mulgt0sr  6819  mulextsr1lem  6821  recidpirqlemcalc  6890
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