Theorem addcan 5274

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 5272	. . 3 |- E.x e. CC (A + x) = 0
3		addcan.2	. . . . . . . . . 10 |- B e. CC
4		axaddass 5200	. . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ B e. CC) -> ((x + A) + B) = (x + (A + B)))
5	3, 4	mp3an3 901	. . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x + A) + B) = (x + (A + B)))
6		addcan.3	. . . . . . . . . 10 |- C e. CC
7		axaddass 5200	. . . . . . . . . 10 |- ((x e. CC /\ A e. CC /\ C e. CC) -> ((x + A) + C) = (x + (A + C)))
8	6, 7	mp3an3 901	. . . . . . . . 9 |- ((x e. CC /\ A e. CC) -> ((x + A) + C) = (x + (A + C)))
9	5, 8	eqeq12d 1465	. . . . . . . 8 |- ((x e. CC /\ A e. CC) -> (((x + A) + B) = ((x + A) + C) <-> (x + (A + B)) = (x + (A + C))))
10	1, 9	mpan2 693	. . . . . . 7 |- (x e. CC -> (((x + A) + B) = ((x + A) + C) <-> (x + (A + B)) = (x + (A + C))))
11		opreq2 3908	. . . . . . 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 5198	. . . . . . . . 9 |- ((A e. CC /\ x e. CC) -> (A + x) = (x + A))
15	1, 14	mpan 692	. . . . . . . 8 |- (x e. CC -> (A + x) = (x + A))
16	15	eqeq1d 1459	. . . . . . 7 |- (x e. CC -> ((A + x) = 0 <-> (x + A) = 0))
17		opreq1 3907	. . . . . . . . 9 |- ((x + A) = 0 -> ((x + A) + B) = (0 + B))
18	3	addid2 5254	. . . . . . . . 9 |- (0 + B) = B
19	17, 18	syl6eq 1499	. . . . . . . 8 |- ((x + A) = 0 -> ((x + A) + B) = B)
20		opreq1 3907	. . . . . . . . 9 |- ((x + A) = 0 -> ((x + A) + C) = (0 + C))
21	6	addid2 5254	. . . . . . . . 9 |- (0 + C) = C
22	20, 21	syl6eq 1499	. . . . . . . 8 |- ((x + A) = 0 -> ((x + A) + C) = C)
23	19, 22	eqeq12d 1465	. . . . . . 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 1720	. . 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 3908	. 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 1099   e. wcel 1105  E.wrex 1622  (class class class)co 3902  CCcc 5155  0cc0 5157   + caddc 5160

This theorem is referenced by:  addcant 5275  nn0opth 6547  cru 6618  cjreb 6667  pjnel 9799  lnopunilem1 10064

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-inf2 4549

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-sbc 1913  df-csb 1973  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-pss 2026  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-fv 3161  df-rdg 3871  df-opr 3904  df-oprab 3905  df-1st 4017  df-2nd 4018  df-1o 4071  df-oadd 4073  df-omul 4074  df-er 4199  df-ec 4201  df-qs 4204  df-ni 4923  df-pli 4924  df-mi 4925  df-lti 4926  df-plpq 4958  df-mpq 4959  df-enq 4960  df-nq 4961  df-plq 4962  df-mq 4963  df-rq 4964  df-ltq 4965  df-1q 4966  df-np 5009  df-1p 5010  df-plp 5011  df-mp 5012  df-ltp 5013  df-plpr 5087  df-mpr 5088  df-enr 5089  df-nr 5090  df-plr 5091  df-mr 5092  df-0r 5094  df-1r 5095  df-m1r 5096  df-c 5163  df-0 5164  df-1 5165  df-i 5166  df-r 5167  df-plus 5168  df-mul 5169

Copyright terms: Public domain