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Theorem mulcomprg 7640
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 7535 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 elprnql 7541 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (1st𝐵)) → 𝑧Q)
31, 2sylan 283 . . . . . . . 8 ((𝐵P𝑧 ∈ (1st𝐵)) → 𝑧Q)
4 prop 7535 . . . . . . . . . . . . 13 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
5 elprnql 7541 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
64, 5sylan 283 . . . . . . . . . . . 12 ((𝐴P𝑦 ∈ (1st𝐴)) → 𝑦Q)
7 mulcomnqg 7443 . . . . . . . . . . . . 13 ((𝑧Q𝑦Q) → (𝑧 ·Q 𝑦) = (𝑦 ·Q 𝑧))
87eqeq2d 2205 . . . . . . . . . . . 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 2491 . . . . . . . . 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 2491 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑦 ·Q 𝑧)))
16 rexcom 2658 . . . . 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 2749 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (1st𝐵)∃𝑦 ∈ (1st𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐵)𝑥 = (𝑦 ·Q 𝑧)})
19 elprnqu 7542 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
201, 19sylan 283 . . . . . . . 8 ((𝐵P𝑧 ∈ (2nd𝐵)) → 𝑧Q)
21 elprnqu 7542 . . . . . . . . . . . . 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 2491 . . . . . . . . 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 2491 . . . . 5 ((𝐴P𝐵P) → (∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑦 ·Q 𝑧)))
30 rexcom 2658 . . . . 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 2749 . . 3 ((𝐴P𝐵P) → {𝑥Q ∣ ∃𝑧 ∈ (2nd𝐵)∃𝑦 ∈ (2nd𝐴)𝑥 = (𝑧 ·Q 𝑦)} = {𝑥Q ∣ ∃𝑦 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐵)𝑥 = (𝑦 ·Q 𝑧)})
3318, 32opeq12d 3812 . 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 7599 . . 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 7599 . 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 2237 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 2164  wrex 2473  {crab 2476  cop 3621  cfv 5254  (class class class)co 5918  1st c1st 6191  2nd c2nd 6192  Qcnq 7340   ·Q cmq 7343  Pcnp 7351   ·P cmp 7354
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-mqqs 7410  df-inp 7526  df-imp 7529
This theorem is referenced by:  ltmprr  7702  mulcmpblnrlemg  7800  mulcomsrg  7817  mulasssrg  7818  m1m1sr  7821  recexgt0sr  7833  mulgt0sr  7838  mulextsr1lem  7840  recidpirqlemcalc  7917
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