Theorem addcan 5323

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
addcan	|- ((A + B) = (A + C) <-> B = C)

Proof of Theorem addcan

Step	Hyp	Ref	Expression
1		addcan.1	. . . 4 |- A e. CC
2	1	cnegex 5321	. . 3 |- E.x e. CC (A + x) = 0
3		addcan.2	. . . . . . . . . 10 |- B e. CC
4		axaddass 5249	. . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ B e. CC) -> ((x + A) + B) = (x + (A + B)))
5	3, 4	mp3an3 902	. . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x + A) + B) = (x + (A + B)))
6		addcan.3	. . . . . . . . . 10 |- C e. CC
7		axaddass 5249	. . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ C e. CC) -> ((x + A) + C) = (x + (A + C)))
8	6, 7	mp3an3 902	. . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x + A) + C) = (x + (A + C)))
9	5, 8	eqeq12d 1481	. . . . . . . 8 |- ((x e. CC /\ A e. CC) -> (((x + A) + B) = ((x + A) + C) <-> (x + (A + B)) = (x + (A + C))))
10	1, 9	mpan2 694	. . . . . . 7 |- (x e. CC -> (((x + A) + B) = ((x + A) + C) <-> (x + (A + B)) = (x + (A + C))))
11		opreq2 3954	. . . . . . 7 |- ((A + B) = (A + C) -> (x + (A + B)) = (x + (A + C)))
12	10, 11	syl5bir 210	. . . . . 6 |- (x e. CC -> ((A + B) = (A + C) -> ((x + A) + B) = ((x + A) + C)))
13	12	adantr 389	. . . . 5 |- ((x e. CC /\ (A + x) = 0) -> ((A + B) = (A + C) -> ((x + A) + B) = ((x + A) + C)))
14		axaddcom 5247	. . . . . . . . 9 |- ((A e. CC /\ x e. CC) -> (A + x) = (x + A))
15	1, 14	mpan 693	. . . . . . . 8 |- (x e. CC -> (A + x) = (x + A))
16	15	eqeq1d 1475	. . . . . . 7 |- (x e. CC -> ((A + x) = 0 <-> (x + A) = 0))
17		opreq1 3953	. . . . . . . . 9 |- ((x + A) = 0 -> ((x + A) + B) = (0 + B))
18	3	addid2 5303	. . . . . . . . 9 |- (0 + B) = B
19	17, 18	syl6eq 1515	. . . . . . . 8 |- ((x + A) = 0 -> ((x + A) + B) = B)
20		opreq1 3953	. . . . . . . . 9 |- ((x + A) = 0 -> ((x + A) + C) = (0 + C))
21	6	addid2 5303	. . . . . . . . 9 |- (0 + C) = C
22	20, 21	syl6eq 1515	. . . . . . . 8 |- ((x + A) = 0 -> ((x + A) + C) = C)
23	19, 22	eqeq12d 1481	. . . . . . 7 |- ((x + A) = 0 -> (((x + A) + B) = ((x + A) + C) <-> B = C))
24	16, 23	syl6bi 214	. . . . . 6 |- (x e. CC -> ((A + x) = 0 -> (((x + A) + B) = ((x + A) + C) <-> B = C)))
25	24	imp 350	. . . . 5 |- ((x e. CC /\ (A + x) = 0) -> (((x + A) + B) = ((x + A) + C) <-> B = C))
26	13, 25	sylibd 202	. . . 4 |- ((x e. CC /\ (A + x) = 0) -> ((A + B) = (A + C) -> B = C))
27	26	r19.23aiva 1736	. . 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 3954	. 2 |- (B = C -> (A + B) = (A + C))
30	28, 29	impbi 157	1 |- ((A + B) = (A + C) <-> B = C)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wrex 1638  (class class class)co 3948  CCcc 5204  0cc0 5206   + caddc 5209

This theorem is referenced by:  addcant 5324  nn0opth 6596  cru 6667  cjreb 6716  pjnel 9585  lnopunilem1 9850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218

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