Theorem mul02lem2 10213

Description: Lemma for mul02 10214. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
mul02lem2	|- ( A e. RR -> ( 0 x. A ) = 0 )

Proof of Theorem mul02lem2

Step	Hyp	Ref	Expression
1		ax-1ne0 10005	. 2 |- 1 =/= 0
2		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
3		mul02lem1 10212	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ 1 e. CC ) -> 1 = ( 1 + 1 ) )
4	2, 3	mpan2 707	. . . . . . . 8 |- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> 1 = ( 1 + 1 ) )
5	4	eqcomd 2628	. . . . . . 7 |- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> ( 1 + 1 ) = 1 )
6	5	oveq2d 6666	. . . . . 6 |- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> ( ( _i x. _i ) + ( 1 + 1 ) ) = ( ( _i x. _i ) + 1 ) )
7		ax-icn 9995	. . . . . . . . 9 |- _i e. CC
8	7, 7	mulcli 10045	. . . . . . . 8 |- ( _i x. _i ) e. CC
9	8, 2, 2	addassi 10048	. . . . . . 7 |- ( ( ( _i x. _i ) + 1 ) + 1 ) = ( ( _i x. _i ) + ( 1 + 1 ) )
10		ax-i2m1 10004	. . . . . . . 8 |- ( ( _i x. _i ) + 1 ) = 0
11	10	oveq1i 6660	. . . . . . 7 |- ( ( ( _i x. _i ) + 1 ) + 1 ) = ( 0 + 1 )
12	9, 11	eqtr3i 2646	. . . . . 6 |- ( ( _i x. _i ) + ( 1 + 1 ) ) = ( 0 + 1 )
13		00id 10211	. . . . . . 7 |- ( 0 + 0 ) = 0
14	10, 13	eqtr4i 2647	. . . . . 6 |- ( ( _i x. _i ) + 1 ) = ( 0 + 0 )
15	6, 12, 14	3eqtr3g 2679	. . . . 5 |- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> ( 0 + 1 ) = ( 0 + 0 ) )
16		1re 10039	. . . . . 6 |- 1 e. RR
17		0re 10040	. . . . . 6 |- 0 e. RR
18		readdcan 10210	. . . . . 6 |- ( ( 1 e. RR /\ 0 e. RR /\ 0 e. RR ) -> ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 ) )
19	16, 17, 17, 18	mp3an 1424	. . . . 5 |- ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 )
20	15, 19	sylib 208	. . . 4 |- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> 1 = 0 )
21	20	ex 450	. . 3 |- ( A e. RR -> ( ( 0 x. A ) =/= 0 -> 1 = 0 ) )
22	21	necon1d 2816	. 2 |- ( A e. RR -> ( 1 =/= 0 -> ( 0 x. A ) = 0 ) )
23	1, 22	mpi 20	1 |- ( A e. RR -> ( 0 x. A ) = 0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + 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:  mul02  10214  rexmul  12101  mbfmulc2lem  23414  i1fmulc  23470  itg1mulc  23471  stoweidlem34  40251  ztprmneprm  42125  nn0sumshdiglemA  42413  nn0sumshdiglem1  42415