Theorem mulcomprg 7947

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 7842	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7848	. . . . . . . . 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 7842	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7848	. . . . . . . . . . . . 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 7750	. . . . . . . . . . . . 13 |- ( ( z e. Q. /\ y e. Q. ) -> ( z .Q y ) = ( y .Q z ) )
8	7	eqeq2d 2250	. . . . . . . . . . . 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 404	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 1st ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
11	10	rexbidva 2547	. . . . . . . . 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 404	. . . . . 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 2547	. . . . 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 2715	. . . . 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 2810	. . 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 7849	. . . . . . . . 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 7849	. . . . . . . . . . . . 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 404	. . . . . . . . . 10 |- ( ( ( z e. Q. /\ A e. P. ) /\ y e. ( 2nd ` A ) ) -> ( x = ( z .Q y ) <-> x = ( y .Q z ) ) )
25	24	rexbidva 2547	. . . . . . . . 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 404	. . . . . 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 2547	. . . . 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 2715	. . . . 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 2810	. . 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 3912	. 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 7906	. . 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 7906	. 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 2282	1 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3712   ` cfv 5377  (class class class)co 6085   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   .Q cmq 7650   P.cnp 7658   .P. cmp 7661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-mi 7673  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-mqqs 7717  df-inp 7833  df-imp 7836

This theorem is used by:  ltmprr  8009  mulcmpblnrlemg  8107  mulcomsrg  8124  mulasssrg  8125  m1m1sr  8128  recexgt0sr  8140  mulgt0sr  8145  mulextsr1lem  8147  recidpirqlemcalc  8224