Theorem mulcomprg 6770

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 6665	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 6671	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ z e. ( 1st ` B ) ) -> z e. Q. )
3	1, 2	sylan 277	. . . . . . . 8 |- ( ( B e. P. /\ z e. ( 1st ` B ) ) -> z e. Q. )
4		prop 6665	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 6671	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
6	4, 5	sylan 277	. . . . . . . . . . . 12 |- ( ( A e. P. /\ y e. ( 1st ` A ) ) -> y e. Q. )
7		mulcomnqg 6573	. . . . . . . . . . . . 13 |- ( ( z e. Q. /\ y e. Q. ) -> ( z .Q y ) = ( y .Q z ) )
8	7	eqeq2d 2092	. . . . . . . . . . . 12 |- ( ( z e. Q. /\ y e. Q. ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
9	6, 8	sylan2 280	. . . . . . . . . . 11 |- ( ( z e. Q. /\ ( A e. P. /\ y e. ( 1st ` A ) ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
10	9	anassrs 392	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 1st ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
11	10	rexbidva 2365	. . . . . . . . 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 264	. . . . . . . 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 280	. . . . . . 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 392	. . . . . 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 2365	. . . . 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 2518	. . . . 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	syl6bb 194	. . . 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 2593	. . 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 6672	. . . . . . . . 9 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ z e. ( 2nd ` B ) ) -> z e. Q. )
20	1, 19	sylan 277	. . . . . . . 8 |- ( ( B e. P. /\ z e. ( 2nd ` B ) ) -> z e. Q. )
21		elprnqu 6672	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
22	4, 21	sylan 277	. . . . . . . . . . . 12 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
23	22, 8	sylan2 280	. . . . . . . . . . 11 |- ( ( z e. Q. /\ ( A e. P. /\ y e. ( 2nd ` A ) ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
24	23	anassrs 392	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 2nd ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
25	24	rexbidva 2365	. . . . . . . . 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 264	. . . . . . . 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 280	. . . . . . 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 392	. . . . . 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 2365	. . . . 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 2518	. . . . 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	syl6bb 194	. . . 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 2593	. . 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 3578	. 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 6729	. . 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 264	. 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 6729	. 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 2124	1 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   {crab 2352   <.cop 3401   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   .Q cmq 6473   P.cnp 6481   .P. cmp 6484

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-mqqs 6540  df-inp 6656  df-imp 6659

This theorem is referenced by:  ltmprr  6832  mulcmpblnrlemg  6917  mulcomsrg  6934  mulasssrg  6935  m1m1sr  6938  recexgt0sr  6950  mulgt0sr  6954  mulextsr1lem  6956  recidpirqlemcalc  7025