Theorem mulcompr 9845

Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
mulcompr	|- ( A .P. B ) = ( B .P. A )

Proof of Theorem mulcompr

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

Step	Hyp	Ref	Expression
1		mpv 9833	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = { x | E. z e. A E. y e. B x = ( z .Q y ) } )
2		mpv 9833	. . . . 5 |- ( ( B e. P. /\ A e. P. ) -> ( B .P. A ) = { x | E. y e. B E. z e. A x = ( y .Q z ) } )
3		mulcomnq 9775	. . . . . . . . 9 |- ( y .Q z ) = ( z .Q y )
4	3	eqeq2i 2634	. . . . . . . 8 |- ( x = ( y .Q z ) <-> x = ( z .Q y ) )
5	4	2rexbii 3042	. . . . . . 7 |- ( E. y e. B E. z e. A x = ( y .Q z ) <-> E. y e. B E. z e. A x = ( z .Q y ) )
6		rexcom 3099	. . . . . . 7 |- ( E. y e. B E. z e. A x = ( z .Q y ) <-> E. z e. A E. y e. B x = ( z .Q y ) )
7	5, 6	bitri 264	. . . . . 6 |- ( E. y e. B E. z e. A x = ( y .Q z ) <-> E. z e. A E. y e. B x = ( z .Q y ) )
8	7	abbii 2739	. . . . 5 |- { x | E. y e. B E. z e. A x = ( y .Q z ) } = { x | E. z e. A E. y e. B x = ( z .Q y ) }
9	2, 8	syl6eq 2672	. . . 4 |- ( ( B e. P. /\ A e. P. ) -> ( B .P. A ) = { x | E. z e. A E. y e. B x = ( z .Q y ) } )
10	9	ancoms 469	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( B .P. A ) = { x | E. z e. A E. y e. B x = ( z .Q y ) } )
11	1, 10	eqtr4d 2659	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )
12		dmmp 9835	. . 3 |- dom .P. = ( P. X. P. )
13	12	ndmovcom 6821	. 2 |- ( -. ( A e. P. /\ B e. P. ) -> ( A .P. B ) = ( B .P. A ) )
14	11, 13	pm2.61i 176	1 |- ( A .P. B ) = ( B .P. A )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913  (class class class)co 6650   .Q cmq 9678   P.cnp 9681   .P. cmp 9684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-mpq 9731  df-enq 9733  df-nq 9734  df-erq 9735  df-mq 9737  df-1nq 9738  df-np 9803  df-mp 9806

This theorem is referenced by:  mulcmpblnrlem  9891  mulcomsr  9910  mulasssr  9911  m1m1sr  9914  recexsrlem  9924  mulgt0sr  9926