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Theorem mul02lem1 10212
Description: Lemma for mul02 10214. If any real does not produce 0 when multiplied by 0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
mul02lem1 |- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) -> B = ( B + B ) )
Proof of Theorem mul02lem1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 0re 10040 . . . . 5 |- 0 e. RR
2 remulcl 10021 . . . . 5 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 x. A ) e. RR )
31, 2mpan 706 . . . 4 |- ( A e. RR -> ( 0 x. A ) e. RR )
4 ax-rrecex 10008 . . . 4 |- ( ( ( 0 x. A ) e. RR /\ ( 0 x. A ) =/= 0 ) -> E. y e. RR ( ( 0 x. A ) x. y ) = 1 )
53, 4sylan 488 . . 3 |- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> E. y e. RR ( ( 0 x. A ) x. y ) = 1 )
65adantr 481 . 2 |- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) -> E. y e. RR ( ( 0 x. A ) x. y ) = 1 )
7 00id 10211 . . . . 5 |- ( 0 + 0 ) = 0
87oveq2i 6661 . . . 4 |- ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) ) = ( ( ( y x. A ) x. B ) x. 0 )
98eqcomi 2631 . . 3 |- ( ( ( y x. A ) x. B ) x. 0 ) = ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) )
10 simprl 794 . . . . . . 7 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> y e. RR )
1110recnd 10068 . . . . . 6 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> y e. CC )
12 simplll 798 . . . . . . 7 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> A e. RR )
1312recnd 10068 . . . . . 6 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> A e. CC )
1411, 13mulcld 10060 . . . . 5 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( y x. A ) e. CC )
15 simplr 792 . . . . 5 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> B e. CC )
16 0cn 10032 . . . . . 6 |- 0 e. CC
17 mul32 10203 . . . . . 6 |- ( ( ( y x. A ) e. CC /\ B e. CC /\ 0 e. CC ) -> ( ( ( y x. A ) x. B ) x. 0 ) = ( ( ( y x. A ) x. 0 ) x. B ) )
1816, 17mp3an3 1413 . . . . 5 |- ( ( ( y x. A ) e. CC /\ B e. CC ) -> ( ( ( y x. A ) x. B ) x. 0 ) = ( ( ( y x. A ) x. 0 ) x. B ) )
1914, 15, 18syl2anc 693 . . . 4 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( ( y x. A ) x. B ) x. 0 ) = ( ( ( y x. A ) x. 0 ) x. B ) )
20 mul31 10204 . . . . . . . . 9 |- ( ( y e. CC /\ A e. CC /\ 0 e. CC ) -> ( ( y x. A ) x. 0 ) = ( ( 0 x. A ) x. y ) )
2116, 20mp3an3 1413 . . . . . . . 8 |- ( ( y e. CC /\ A e. CC ) -> ( ( y x. A ) x. 0 ) = ( ( 0 x. A ) x. y ) )
2211, 13, 21syl2anc 693 . . . . . . 7 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( y x. A ) x. 0 ) = ( ( 0 x. A ) x. y ) )
23 simprr 796 . . . . . . 7 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( 0 x. A ) x. y ) = 1 )
2422, 23eqtrd 2656 . . . . . 6 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( y x. A ) x. 0 ) = 1 )
2524oveq1d 6665 . . . . 5 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( ( y x. A ) x. 0 ) x. B ) = ( 1 x. B ) )
26 mulid2 10038 . . . . . 6 |- ( B e. CC -> ( 1 x. B ) = B )
2726ad2antlr 763 . . . . 5 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( 1 x. B ) = B )
2825, 27eqtrd 2656 . . . 4 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( ( y x. A ) x. 0 ) x. B ) = B )
2919, 28eqtrd 2656 . . 3 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( ( y x. A ) x. B ) x. 0 ) = B )
3014, 15mulcld 10060 . . . . 5 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( y x. A ) x. B ) e. CC )
31 adddi 10025 . . . . . 6 |- ( ( ( ( y x. A ) x. B ) e. CC /\ 0 e. CC /\ 0 e. CC ) -> ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) ) = ( ( ( ( y x. A ) x. B ) x. 0 ) + ( ( ( y x. A ) x. B ) x. 0 ) ) )
3216, 16, 31mp3an23 1416 . . . . 5 |- ( ( ( y x. A ) x. B ) e. CC -> ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) ) = ( ( ( ( y x. A ) x. B ) x. 0 ) + ( ( ( y x. A ) x. B ) x. 0 ) ) )
3330, 32syl 17 . . . 4 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) ) = ( ( ( ( y x. A ) x. B ) x. 0 ) + ( ( ( y x. A ) x. B ) x. 0 ) ) )
3429, 29oveq12d 6668 . . . 4 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( ( ( y x. A ) x. B ) x. 0 ) + ( ( ( y x. A ) x. B ) x. 0 ) ) = ( B + B ) )
3533, 34eqtrd 2656 . . 3 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) ) = ( B + B ) )
369, 29, 353eqtr3a 2680 . 2 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> B = ( B + B ) )
376, 36rexlimddv 3035 1 |- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) -> B = ( B + B ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941
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-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012
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-nel 2898  df-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079
This theorem is referenced by:  mul02lem2  10213
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