Theorem mulcompr 10790

Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcompr	⊢ (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)

Proof of Theorem mulcompr

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpv 10778	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)})
2		mpv 10778	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧)})
3		mulcomnq 10720	. . . . . . . . 9 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
4	3	eqeq2i 2753	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ·Q 𝑧) ↔ 𝑥 = (𝑧 ·Q 𝑦))
5	4	2rexbii 3181	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 ·Q 𝑦))
6		rexcom 3284	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 ·Q 𝑦) ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦))
7	5, 6	bitri 274	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧) ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦))
8	7	abbii 2810	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑦 ·Q 𝑧)} = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)}
9	2, 8	eqtrdi 2796	. . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)})
10	9	ancoms 459	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐵 ·P 𝐴) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 = (𝑧 ·Q 𝑦)})
11	1, 10	eqtr4d 2783	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
12		dmmp 10780	. . 3 ⊢ dom ·P = (P × P)
13	12	ndmovcom 7454	. 2 ⊢ (¬ (𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴))
14	11, 13	pm2.61i 182	1 ⊢ (𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)

Syntax hints:   ∧ wa 396   = wceq 1542   ∈ wcel 2110  {cab 2717  ∃wrex 3067  (class class class)co 7272   ·_Q cmq 10623  Pcnp 10626   ·_P cmp 10629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7583  ax-inf2 9387

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7275  df-oprab 7276  df-mpo 7277  df-om 7708  df-1st 7825  df-2nd 7826  df-frecs 8089  df-wrecs 8120  df-recs 8194  df-rdg 8233  df-1o 8289  df-oadd 8293  df-omul 8294  df-er 8490  df-ni 10639  df-mi 10641  df-lti 10642  df-mpq 10676  df-enq 10678  df-nq 10679  df-erq 10680  df-mq 10682  df-1nq 10683  df-np 10748  df-mp 10751

This theorem is referenced by:  mulcmpblnrlem  10837  mulcomsr  10856  mulasssr  10857  m1m1sr  10860  recexsrlem  10870  mulgt0sr  10872