Theorem addcaniOLD 6506

Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18.

Hypotheses

Ref	Expression
addcan.1	|- A e. CC
addcan.2	|- B e. CC
addcan.3	|- C e. CC

Assertion

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

Proof of Theorem addcaniOLD

Step	Hyp	Ref	Expression
1		addcan.1	. . . 4 |- A e. CC
2	1	cnegexi 6503	. . 3 |- E.x e. CC (A + x) = 0
3		addcan.2	. . . . . . . . . 10 |- B e. CC
4		addass 6460	. . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ B e. CC) -> ((x + A) + B) = (x + (A + B)))
5	3, 4	mp3an3 1180	. . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x + A) + B) = (x + (A + B)))
6		addcan.3	. . . . . . . . . 10 |- C e. CC
7		addass 6460	. . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ C e. CC) -> ((x + A) + C) = (x + (A + C)))
8	6, 7	mp3an3 1180	. . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x + A) + C) = (x + (A + C)))
9	5, 8	eqeq12d 1899	. . . . . . . 8 |- ((x e. CC /\ A e. CC) -> (((x + A) + B) = ((x + A) + C) <-> (x + (A + B)) = (x + (A + C))))
10	1, 9	mpan2 760	. . . . . . 7 |- (x e. CC -> (((x + A) + B) = ((x + A) + C) <-> (x + (A + B)) = (x + (A + C))))
11		opreq2 4890	. . . . . . 7 |- ((A + B) = (A + C) -> (x + (A + B)) = (x + (A + C)))
12	10, 11	syl5bir 227	. . . . . 6 |- (x e. CC -> ((A + B) = (A + C) -> ((x + A) + B) = ((x + A) + C)))
13	12	adantr 425	. . . . 5 |- ((x e. CC /\ (A + x) = 0) -> ((A + B) = (A + C) -> ((x + A) + B) = ((x + A) + C)))
14		addcom 6458	. . . . . . . . 9 |- ((A e. CC /\ x e. CC) -> (A + x) = (x + A))
15	1, 14	mpan 759	. . . . . . . 8 |- (x e. CC -> (A + x) = (x + A))
16	15	eqeq1d 1892	. . . . . . 7 |- (x e. CC -> ((A + x) = 0 <-> (x + A) = 0))
17		opreq1 4889	. . . . . . . . 9 |- ((x + A) = 0 -> ((x + A) + B) = (0 + B))
18	3	addid2i 6484	. . . . . . . . 9 |- (0 + B) = B
19	17, 18	syl6eq 1944	. . . . . . . 8 |- ((x + A) = 0 -> ((x + A) + B) = B)
20		opreq1 4889	. . . . . . . . 9 |- ((x + A) = 0 -> ((x + A) + C) = (0 + C))
21	6	addid2i 6484	. . . . . . . . 9 |- (0 + C) = C
22	20, 21	syl6eq 1944	. . . . . . . 8 |- ((x + A) = 0 -> ((x + A) + C) = C)
23	19, 22	eqeq12d 1899	. . . . . . 7 |- ((x + A) = 0 -> (((x + A) + B) = ((x + A) + C) <-> B = C))
24	16, 23	syl6bi 231	. . . . . 6 |- (x e. CC -> ((A + x) = 0 -> (((x + A) + B) = ((x + A) + C) <-> B = C)))
25	24	imp 377	. . . . 5 |- ((x e. CC /\ (A + x) = 0) -> (((x + A) + B) = ((x + A) + C) <-> B = C))
26	13, 25	sylibd 219	. . . 4 |- ((x e. CC /\ (A + x) = 0) -> ((A + B) = (A + C) -> B = C))
27	26	r19.23aiva 2212	. . 3 |- (E.x e. CC (A + x) = 0 -> ((A + B) = (A + C) -> B = C))
28	2, 27	ax-mp 7	. 2 |- ((A + B) = (A + C) -> B = C)
29		opreq2 4890	. 2 |- (B = C -> (A + B) = (A + C))
30	28, 29	impbii 174	1 |- ((A + B) = (A + C) <-> B = C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  (class class class)co 4884  CCcc 6384  0cc0 6386   + caddc 6389

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398

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