Theorem mulcomprg 7502

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	|- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )

Proof of Theorem mulcomprg

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

Step	Hyp	Ref	Expression
1		prop 7397	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7403	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ z e. ( 1st ` B ) ) -> z e. Q. )
3	1, 2	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ z e. ( 1st ` B ) ) -> z e. Q. )
4		prop 7397	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7403	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
6	4, 5	sylan 281	. . . . . . . . . . . 12 |- ( ( A e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
7		mulcomnqg 7305	. . . . . . . . . . . . 13 |- ( ( z e. Q. /\ y e. Q. ) -> ( z .Q y ) = ( y .Q z ) )
8	7	eqeq2d 2169	. . . . . . . . . . . 12 |- ( ( z e. Q. /\ y e. Q. ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
9	6, 8	sylan2 284	. . . . . . . . . . 11 |- ( ( z e. Q. /\ ( A e. P. /\ y e. ( 1st ` A ) ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
10	9	anassrs 398	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 1st ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
11	10	rexbidva 2454	. . . . . . . . 9 |- ( ( z e. Q. /\ A e. P. ) -> ( E. y e. ( 1st ` A ) x = ( z .Q y ) <-> E. y e. ( 1st ` A ) x = ( y .Q z ) ) )
12	11	ancoms 266	. . . . . . . 8 |- ( ( A e. P. /\ z e. Q. ) -> ( E. y e. ( 1st ` A ) x = ( z .Q y ) <-> E. y e. ( 1st ` A ) x = ( y .Q z ) ) )
13	3, 12	sylan2 284	. . . . . . 7 |- ( ( A e. P. /\ ( B e. P. /\ z e. ( 1st ` B ) ) ) -> ( E. y e. ( 1st ` A ) x = ( z .Q y ) <-> E. y e. ( 1st ` A ) x = ( y .Q z ) ) )
14	13	anassrs 398	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ z e. ( 1st ` B ) ) -> ( E. y e. ( 1st ` A ) x = ( z .Q y ) <-> E. y e. ( 1st ` A ) x = ( y .Q z ) ) )
15	14	rexbidva 2454	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( z .Q y ) <-> E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( y .Q z ) ) )
16		rexcom 2621	. . . . 5 |- ( E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( y .Q z ) <-> E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) )
17	15, 16	bitrdi 195	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( z .Q y ) <-> E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) ) )
18	17	rabbidv 2701	. . 3 |- ( ( A e. P. /\ B e. P. ) -> { x e. Q. | E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( z .Q y ) } = { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) } )
19		elprnqu 7404	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ z e. ( 2nd ` B ) ) -> z e. Q. )
20	1, 19	sylan 281	. . . . . . . 8 |- ( ( B e. P. /\ z e. ( 2nd ` B ) ) -> z e. Q. )
21		elprnqu 7404	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
22	4, 21	sylan 281	. . . . . . . . . . . 12 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
23	22, 8	sylan2 284	. . . . . . . . . . 11 |- ( ( z e. Q. /\ ( A e. P. /\ y e. ( 2nd ` A ) ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
24	23	anassrs 398	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 2nd ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
25	24	rexbidva 2454	. . . . . . . . 9 |- ( ( z e. Q. /\ A e. P. ) -> ( E. y e. ( 2nd ` A ) x = ( z .Q y ) <-> E. y e. ( 2nd ` A ) x = ( y .Q z ) ) )
26	25	ancoms 266	. . . . . . . 8 |- ( ( A e. P. /\ z e. Q. ) -> ( E. y e. ( 2nd ` A ) x = ( z .Q y ) <-> E. y e. ( 2nd ` A ) x = ( y .Q z ) ) )
27	20, 26	sylan2 284	. . . . . . 7 |- ( ( A e. P. /\ ( B e. P. /\ z e. ( 2nd ` B ) ) ) -> ( E. y e. ( 2nd ` A ) x = ( z .Q y ) <-> E. y e. ( 2nd ` A ) x = ( y .Q z ) ) )
28	27	anassrs 398	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ z e. ( 2nd ` B ) ) -> ( E. y e. ( 2nd ` A ) x = ( z .Q y ) <-> E. y e. ( 2nd ` A ) x = ( y .Q z ) ) )
29	28	rexbidva 2454	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( z .Q y ) <-> E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( y .Q z ) ) )
30		rexcom 2621	. . . . 5 |- ( E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( y .Q z ) <-> E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) )
31	29, 30	bitrdi 195	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( z .Q y ) <-> E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) ) )
32	31	rabbidv 2701	. . 3 |- ( ( A e. P. /\ B e. P. ) -> { x e. Q. | E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( z .Q y ) } = { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) } )
33	18, 32	opeq12d 3751	. 2 |- ( ( A e. P. /\ B e. P. ) -> <. { x e. Q. | E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( z .Q y ) } , { x e. Q. | E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( z .Q y ) } >. = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) } >. )
34		mpvlu 7461	. . 3 |- ( ( B e. P. /\ A e. P. ) -> ( B .P. A ) = <. { x e. Q. | E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( z .Q y ) } , { x e. Q. | E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( z .Q y ) } >. )
35	34	ancoms 266	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( B .P. A ) = <. { x e. Q. | E. z e. ( 1st ` B ) E. y e. ( 1st ` A ) x = ( z .Q y ) } , { x e. Q. | E. z e. ( 2nd ` B ) E. y e. ( 2nd ` A ) x = ( z .Q y ) } >. )
36		mpvlu 7461	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y .Q z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y .Q z ) } >. )
37	33, 35, 36	3eqtr4rd 2201	1 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436   {crab 2439   <.cop 3564   ` cfv 5172  (class class class)co 5826   1stc1st 6088   2ndc2nd 6089   Q.cnq 7202   .Q cmq 7205   P.cnp 7213   .P. cmp 7216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-mi 7228  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-mqqs 7272  df-inp 7388  df-imp 7391

This theorem is referenced by:  ltmprr  7564  mulcmpblnrlemg  7662  mulcomsrg  7679  mulasssrg  7680  m1m1sr  7683  recexgt0sr  7695  mulgt0sr  7700  mulextsr1lem  7702  recidpirqlemcalc  7779