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Theorem mul02lem1 9767
Description: Lemma for mul02 9769. 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 9608 . . . . 5 |- 0 e. RR
2 remulcl 9589 . . . . 5 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 x. A ) e. RR )
31, 2mpan 670 . . . 4 |- ( A e. RR -> ( 0 x. A ) e. RR )
4 ax-rrecex 9576 . . . 4 |- ( ( ( 0 x. A ) e. RR /\ ( 0 x. A ) =/= 0 ) -> E. y e. RR ( ( 0 x. A ) x. y ) = 1 )
53, 4sylan 471 . . 3 |- ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) -> E. y e. RR ( ( 0 x. A ) x. y ) = 1 )
65adantr 465 . 2 |- ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) -> E. y e. RR ( ( 0 x. A ) x. y ) = 1 )
7 00id 9766 . . . . 5 |- ( 0 + 0 ) = 0
87oveq2i 6306 . . . 4 |- ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) ) = ( ( ( y x. A ) x. B ) x. 0 )
98eqcomi 2480 . . 3 |- ( ( ( y x. A ) x. B ) x. 0 ) = ( ( ( y x. A ) x. B ) x. ( 0 + 0 ) )
10 simprl 755 . . . . . . 7 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> y e. RR )
1110recnd 9634 . . . . . 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 757 . . . . . . 7 |- ( ( ( ( A e. RR /\ ( 0 x. A ) =/= 0 ) /\ B e. CC ) /\ ( y e. RR /\ ( ( 0 x. A ) x. y ) = 1 ) ) -> A e. RR )
1312recnd 9634 . . . . . 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 9628 . . . . 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 754 . . . . 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 9600 . . . . . 6 |- 0 e. CC
17 mul32 9758 . . . . . 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 1313 . . . . 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 661 . . . 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 9759 . . . . . . . . 9 |- ( ( y e. CC /\ A e. CC /\ 0 e. CC ) -> ( ( y x. A ) x. 0 ) = ( ( 0 x. A ) x. y ) )
2116, 20mp3an3 1313 . . . . . . . 8 |- ( ( y e. CC /\ A e. CC ) -> ( ( y x. A ) x. 0 ) = ( ( 0 x. A ) x. y ) )
2211, 13, 21syl2anc 661 . . . . . . 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 756 . . . . . . 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 2508 . . . . . 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 6310 . . . . 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 9606 . . . . . 6 |- ( B e. CC -> ( 1 x. B ) = B )
2726ad2antlr 726 . . . . 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 2508 . . . 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 2508 . . 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 9628 . . . . 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 9593 . . . . . 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 1316 . . . . 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 16 . . . 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 6313 . . . 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 2508 . . 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 2532 . 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 2963 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 369   = wceq 1379   e. wcel 1767   =/= wne 2662   E.wrex 2818  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505   + caddc 9507   x. cmul 9509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-ltxr 9645
This theorem is referenced by:  mul02lem2  9768
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