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Theorem muleqadd 10671
Description: Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)
Assertion
Ref Expression
muleqadd |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) )
Proof of Theorem muleqadd
StepHypRef Expression
1 ax-1cn 9994 . . . . 5 |- 1 e. CC
2 mulsub 10473 . . . . . 6 |- ( ( ( A e. CC /\ 1 e. CC ) /\ ( B e. CC /\ 1 e. CC ) ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
31, 2mpanr2 720 . . . . 5 |- ( ( ( A e. CC /\ 1 e. CC ) /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
41, 3mpanl2 717 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
51mulid1i 10042 . . . . . . 7 |- ( 1 x. 1 ) = 1
65oveq2i 6661 . . . . . 6 |- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 )
76a1i 11 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 ) )
8 mulid1 10037 . . . . . 6 |- ( A e. CC -> ( A x. 1 ) = A )
9 mulid1 10037 . . . . . 6 |- ( B e. CC -> ( B x. 1 ) = B )
108, 9oveqan12d 6669 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
117, 10oveq12d 6668 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) = ( ( ( A x. B ) + 1 ) - ( A + B ) ) )
12 mulcl 10020 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13 addcl 10018 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
14 addsub 10292 . . . . . 6 |- ( ( ( A x. B ) e. CC /\ 1 e. CC /\ ( A + B ) e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
151, 14mp3an2 1412 . . . . 5 |- ( ( ( A x. B ) e. CC /\ ( A + B ) e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
1612, 13, 15syl2anc 693 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
174, 11, 163eqtrd 2660 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
1817eqeq1d 2624 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - 1 ) x. ( B - 1 ) ) = 1 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) )
191addid2i 10224 . . . 4 |- ( 0 + 1 ) = 1
2019eqeq2i 2634 . . 3 |- ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 )
2112, 13subcld 10392 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) - ( A + B ) ) e. CC )
22 0cn 10032 . . . . 5 |- 0 e. CC
23 addcan2 10221 . . . . 5 |- ( ( ( ( A x. B ) - ( A + B ) ) e. CC /\ 0 e. CC /\ 1 e. CC ) -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) )
2422, 1, 23mp3an23 1416 . . . 4 |- ( ( ( A x. B ) - ( A + B ) ) e. CC -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) )
2521, 24syl 17 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) )
2620, 25syl5rbbr 275 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) )
2712, 13subeq0ad 10402 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( A x. B ) = ( A + B ) ) )
2818, 26, 273bitr2rd 297 1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266
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-reu 2919  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269
This theorem is referenced by:  conjmul  10742
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