Theorem mulcomprg 7582

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 7477	. . . . . . . . 9 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		elprnql 7483	. . . . . . . . 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 7477	. . . . . . . . . . . . 13 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
5		elprnql 7483	. . . . . . . . . . . . 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 7385	. . . . . . . . . . . . 13 |- ( ( z e. Q. /\ y e. Q. ) -> ( z .Q y ) = ( y .Q z ) )
8	7	eqeq2d 2189	. . . . . . . . . . . 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 2474	. . . . . . . . 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 2474	. . . . 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 2641	. . . . 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 2728	. . 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 7484	. . . . . . . . 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 7484	. . . . . . . . . . . . 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 2474	. . . . . . . . 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 2474	. . . . 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 2641	. . . . 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 2728	. . 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 3788	. 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 7541	. . 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 7541	. 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 2221	1 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3597   ` cfv 5218  (class class class)co 5878   1stc1st 6142   2ndc2nd 6143   Q.cnq 7282   .Q cmq 7285   P.cnp 7293   .P. cmp 7296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-mi 7308  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-mqqs 7352  df-inp 7468  df-imp 7471

This theorem is referenced by:  ltmprr  7644  mulcmpblnrlemg  7742  mulcomsrg  7759  mulasssrg  7760  m1m1sr  7763  recexgt0sr  7775  mulgt0sr  7780  mulextsr1lem  7782  recidpirqlemcalc  7859