Theorem addcan 4120

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

Hypotheses

Ref	Expression
addcan.1	⊢ A ∈ ℂ
addcan.2	⊢ B ∈ ℂ
addcan.3	⊢ C ∈ ℂ

Assertion

Ref	Expression
addcan	⊢ ((A + B) = (A + C) ↔ B = C)

Proof of Theorem addcan

Step	Hyp	Ref	Expression
1		addcan.1	. . . 4 ⊢ A ∈ ℂ
2	1	negex 4116	. . 3 ⊢ ∃x ∈ ℂ (A + x) = 0
3		addcan.2	. . . . . . . . . . 11 ⊢ B ∈ ℂ
4		axaddass 4072	. . . . . . . . . . 11 ⊢ ((x ∈ ℂ ∧ A ∈ ℂ ∧ B ∈ ℂ) → ((x + A) + B) = (x + (A + B)))
5	3, 4	mp3an3 641	. . . . . . . . . 10 ⊢ ((x ∈ ℂ ∧ A ∈ ℂ) → ((x + A) + B) = (x + (A + B)))
6		addcan.3	. . . . . . . . . . 11 ⊢ C ∈ ℂ
7		axaddass 4072	. . . . . . . . . . 11 ⊢ ((x ∈ ℂ ∧ A ∈ ℂ ∧ C ∈ ℂ) → ((x + A) + C) = (x + (A + C)))
8	6, 7	mp3an3 641	. . . . . . . . . 10 ⊢ ((x ∈ ℂ ∧ A ∈ ℂ) → ((x + A) + C) = (x + (A + C)))
9	5, 8	cleq12d 1115	. . . . . . . . 9 ⊢ ((x ∈ ℂ ∧ A ∈ ℂ) → (((x + A) + B) = ((x + A) + C) ↔ (x + (A + B)) = (x + (A + C))))
10	1, 9	mpan2 519	. . . . . . . 8 ⊢ (x ∈ ℂ → (((x + A) + B) = ((x + A) + C) ↔ (x + (A + B)) = (x + (A + C))))
11		opreq2 3007	. . . . . . . 8 ⊢ ((A + B) = (A + C) → (x + (A + B)) = (x + (A + C)))
12	10, 11	syl5bir 184	. . . . . . 7 ⊢ (x ∈ ℂ → ((A + B) = (A + C) → ((x + A) + B) = ((x + A) + C)))
13	12	adantr 306	. . . . . 6 ⊢ ((x ∈ ℂ ∧ (A + x) = 0) → ((A + B) = (A + C) → ((x + A) + B) = ((x + A) + C)))
14		axaddcom 4070	. . . . . . . . . 10 ⊢ ((A ∈ ℂ ∧ x ∈ ℂ) → (A + x) = (x + A))
15	1, 14	mpan 518	. . . . . . . . 9 ⊢ (x ∈ ℂ → (A + x) = (x + A))
16	15	cleq1d 1109	. . . . . . . 8 ⊢ (x ∈ ℂ → ((A + x) = 0 ↔ (x + A) = 0))
17		opreq1 3006	. . . . . . . . . 10 ⊢ ((x + A) = 0 → ((x + A) + B) = (0 + B))
18	3	addid2 4113	. . . . . . . . . 10 ⊢ (0 + B) = B
19	17, 18	syl6eq 1140	. . . . . . . . 9 ⊢ ((x + A) = 0 → ((x + A) + B) = B)
20		opreq1 3006	. . . . . . . . . 10 ⊢ ((x + A) = 0 → ((x + A) + C) = (0 + C))
21	6	addid2 4113	. . . . . . . . . 10 ⊢ (0 + C) = C
22	20, 21	syl6eq 1140	. . . . . . . . 9 ⊢ ((x + A) = 0 → ((x + A) + C) = C)
23	19, 22	cleq12d 1115	. . . . . . . 8 ⊢ ((x + A) = 0 → (((x + A) + B) = ((x + A) + C) ↔ B = C))
24	16, 23	syl6bi 187	. . . . . . 7 ⊢ (x ∈ ℂ → ((A + x) = 0 → (((x + A) + B) = ((x + A) + C) ↔ B = C)))
25	24	imp 277	. . . . . 6 ⊢ ((x ∈ ℂ ∧ (A + x) = 0) → (((x + A) + B) = ((x + A) + C) ↔ B = C))
26	13, 25	sylibd 177	. . . . 5 ⊢ ((x ∈ ℂ ∧ (A + x) = 0) → ((A + B) = (A + C) → B = C))
27	26	exp 291	. . . 4 ⊢ (x ∈ ℂ → ((A + x) = 0 → ((A + B) = (A + C) → B = C)))
28	27	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ ℂ (A + x) = 0 → ((A + B) = (A + C) → B = C))
29	2, 28	ax-mp 6	. 2 ⊢ ((A + B) = (A + C) → B = C)
30		opreq2 3007	. 2 ⊢ (B = C → (A + B) = (A + C))
31	29, 30	impbi 139	1 ⊢ ((A + B) = (A + C) ↔ B = C)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  (class class class)co 3001  ℂcc 4026  0cc0 4028   + caddc 4031

This theorem is referenced by:  addcan2 4121  addcant 4122  cru 4529  nn0opth 4724  cjre 4811  pjnel 5665

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-r 4038  df-plus 4039

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