Theorem div2sub 10850

Description: Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.)

Assertion

Ref	Expression
div2sub	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC /\ C =/= D ) ) -> ( ( A - B ) / ( C - D ) ) = ( ( B - A ) / ( D - C ) ) )

Proof of Theorem div2sub

Step	Hyp	Ref	Expression
1		subcl 10280	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2		subcl 10280	. . . . 5 |- ( ( C e. CC /\ D e. CC ) -> ( C - D ) e. CC )
3	2	3adant3 1081	. . . 4 |- ( ( C e. CC /\ D e. CC /\ C =/= D ) -> ( C - D ) e. CC )
4		subeq0 10307	. . . . . 6 |- ( ( C e. CC /\ D e. CC ) -> ( ( C - D ) = 0 <-> C = D ) )
5	4	necon3bid 2838	. . . . 5 |- ( ( C e. CC /\ D e. CC ) -> ( ( C - D ) =/= 0 <-> C =/= D ) )
6	5	biimp3ar 1433	. . . 4 |- ( ( C e. CC /\ D e. CC /\ C =/= D ) -> ( C - D ) =/= 0 )
7	3, 6	jca 554	. . 3 |- ( ( C e. CC /\ D e. CC /\ C =/= D ) -> ( ( C - D ) e. CC /\ ( C - D ) =/= 0 ) )
8		div2neg 10748	. . . 4 |- ( ( ( A - B ) e. CC /\ ( C - D ) e. CC /\ ( C - D ) =/= 0 ) -> ( -u ( A - B ) / -u ( C - D ) ) = ( ( A - B ) / ( C - D ) ) )
9	8	3expb 1266	. . 3 |- ( ( ( A - B ) e. CC /\ ( ( C - D ) e. CC /\ ( C - D ) =/= 0 ) ) -> ( -u ( A - B ) / -u ( C - D ) ) = ( ( A - B ) / ( C - D ) ) )
10	1, 7, 9	syl2an 494	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC /\ C =/= D ) ) -> ( -u ( A - B ) / -u ( C - D ) ) = ( ( A - B ) / ( C - D ) ) )
11		negsubdi2 10340	. . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
12		negsubdi2 10340	. . . 4 |- ( ( C e. CC /\ D e. CC ) -> -u ( C - D ) = ( D - C ) )
13	12	3adant3 1081	. . 3 |- ( ( C e. CC /\ D e. CC /\ C =/= D ) -> -u ( C - D ) = ( D - C ) )
14	11, 13	oveqan12d 6669	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC /\ C =/= D ) ) -> ( -u ( A - B ) / -u ( C - D ) ) = ( ( B - A ) / ( D - C ) ) )
15	10, 14	eqtr3d 2658	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC /\ C =/= D ) ) -> ( ( A - B ) / ( C - D ) ) = ( ( B - A ) / ( D - C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   - cmin 10266   -ucneg 10267   / cdiv 10684

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  ax-pre-mulgt0 10013

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-rmo 2920  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-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685

This theorem is referenced by:  div2subd  10851