Theorem mulcomprg 7805

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 7700	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7706	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ z e. ( 1st ` B ) ) -> z e. Q. )
3	1, 2	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ z e. ( 1st ` B ) ) -> z e. Q. )
4		prop 7700	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7706	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
6	4, 5	sylan 283	. . . . . . . . . . . 12 |- ( ( A e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
7		mulcomnqg 7608	. . . . . . . . . . . . 13 |- ( ( z e. Q. /\ y e. Q. ) -> ( z .Q y ) = ( y .Q z ) )
8	7	eqeq2d 2242	. . . . . . . . . . . 12 |- ( ( z e. Q. /\ y e. Q. ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
9	6, 8	sylan2 286	. . . . . . . . . . 11 |- ( ( z e. Q. /\ ( A e. P. /\ y e. ( 1st ` A ) ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
10	9	anassrs 400	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 1st ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
11	10	rexbidva 2528	. . . . . . . . 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 268	. . . . . . . 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 286	. . . . . . 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 400	. . . . . 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 2528	. . . . 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 2696	. . . . 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 196	. . . 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 2790	. . 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 7707	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ z e. ( 2nd ` B ) ) -> z e. Q. )
20	1, 19	sylan 283	. . . . . . . 8 |- ( ( B e. P. /\ z e. ( 2nd ` B ) ) -> z e. Q. )
21		elprnqu 7707	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
22	4, 21	sylan 283	. . . . . . . . . . . 12 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
23	22, 8	sylan2 286	. . . . . . . . . . 11 |- ( ( z e. Q. /\ ( A e. P. /\ y e. ( 2nd ` A ) ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
24	23	anassrs 400	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 2nd ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
25	24	rexbidva 2528	. . . . . . . . 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 268	. . . . . . . 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 286	. . . . . . 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 400	. . . . . 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 2528	. . . . 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 2696	. . . . 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 196	. . . 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 2790	. . 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 3871	. 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 7764	. . 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 268	. 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 7764	. 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 2274	1 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2201   E.wrex 2510   {crab 2513   <.cop 3673   ` cfv 5328  (class class class)co 6023   1stc1st 6306   2ndc2nd 6307   Q.cnq 7505   .Q cmq 7508   P.cnp 7516   .P. cmp 7519

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-irdg 6541  df-oadd 6591  df-omul 6592  df-er 6707  df-ec 6709  df-qs 6713  df-ni 7529  df-mi 7531  df-mpq 7570  df-enq 7572  df-nqqs 7573  df-mqqs 7575  df-inp 7691  df-imp 7694

This theorem is referenced by:  ltmprr  7867  mulcmpblnrlemg  7965  mulcomsrg  7982  mulasssrg  7983  m1m1sr  7986  recexgt0sr  7998  mulgt0sr  8003  mulextsr1lem  8005  recidpirqlemcalc  8082