Theorem remulext2 8770

Description: Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)

Assertion

Ref	Expression
remulext2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C x. A ) # ( C x. B ) -> A # B ) )

Proof of Theorem remulext2

Step	Hyp	Ref	Expression
1		simp1 1021	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2	1	recnd 8198	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
3		simp3 1023	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
4	3	recnd 8198	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
5	2, 4	mulcomd 8191	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A x. C ) = ( C x. A ) )
6		simp2 1022	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
7	6	recnd 8198	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
8	7, 4	mulcomd 8191	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B x. C ) = ( C x. B ) )
9	5, 8	breq12d 4099	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) # ( B x. C ) <-> ( C x. A ) # ( C x. B ) ) )
10		remulext1 8769	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) # ( B x. C ) -> A # B ) )
11	9, 10	sylbird 170	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C x. A ) # ( C x. B ) -> A # B ) )

Syntax hints:   -> wi 4   /\ w3a 1002   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   RRcr 8021   x. cmul 8027   # cap 8751

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752

This theorem is referenced by:  mulext1  8782