Theorem addlsub 10447

Description: Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)

Hypotheses

Ref	Expression
addlsub.a	|- ( ph -> A e. CC )
addlsub.b	|- ( ph -> B e. CC )
addlsub.c	|- ( ph -> C e. CC )

Assertion

Ref	Expression
addlsub	|- ( ph -> ( ( A + B ) = C <-> A = ( C - B ) ) )

Proof of Theorem addlsub

Step	Hyp	Ref	Expression
1		oveq1 6657	. . 3 |- ( ( A + B ) = C -> ( ( A + B ) - B ) = ( C - B ) )
2		addlsub.a	. . . . 5 |- ( ph -> A e. CC )
3		addlsub.b	. . . . 5 |- ( ph -> B e. CC )
4	2, 3	pncand 10393	. . . 4 |- ( ph -> ( ( A + B ) - B ) = A )
5		eqtr2 2642	. . . . . 6 |- ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> ( C - B ) = A )
6	5	eqcomd 2628	. . . . 5 |- ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> A = ( C - B ) )
7	6	a1i 11	. . . 4 |- ( ph -> ( ( ( ( A + B ) - B ) = ( C - B ) /\ ( ( A + B ) - B ) = A ) -> A = ( C - B ) ) )
8	4, 7	mpan2d 710	. . 3 |- ( ph -> ( ( ( A + B ) - B ) = ( C - B ) -> A = ( C - B ) ) )
9	1, 8	syl5 34	. 2 |- ( ph -> ( ( A + B ) = C -> A = ( C - B ) ) )
10		oveq1 6657	. . 3 |- ( A = ( C - B ) -> ( A + B ) = ( ( C - B ) + B ) )
11		addlsub.c	. . . . 5 |- ( ph -> C e. CC )
12	11, 3	npcand 10396	. . . 4 |- ( ph -> ( ( C - B ) + B ) = C )
13		eqtr 2641	. . . . 5 |- ( ( ( A + B ) = ( ( C - B ) + B ) /\ ( ( C - B ) + B ) = C ) -> ( A + B ) = C )
14	13	a1i 11	. . . 4 |- ( ph -> ( ( ( A + B ) = ( ( C - B ) + B ) /\ ( ( C - B ) + B ) = C ) -> ( A + B ) = C ) )
15	12, 14	mpan2d 710	. . 3 |- ( ph -> ( ( A + B ) = ( ( C - B ) + B ) -> ( A + B ) = C ) )
16	10, 15	syl5 34	. 2 |- ( ph -> ( A = ( C - B ) -> ( A + B ) = C ) )
17	9, 16	impbid 202	1 |- ( ph -> ( ( A + B ) = C <-> A = ( C - B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   + caddc 9939   - 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

This theorem is referenced by:  addrsub  10448  subexsub  10449  nn0ob  15100  bj-lineq  33158  blen1b  42382  nn0sumshdiglem1  42415