Theorem addcan 5331

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	cnegex 5329	. . 3 ⊢ ∃x ∈ ℂ (A + x) = 0
3		addcan.2	. . . . . . . . . 10 ⊢ B ∈ ℂ
4		axaddass 5257	. . . . . . . . . 10 ⊢ ((x ∈ ℂ ⋀ A ∈ ℂ ⋀ B ∈ ℂ) → ((x + A) + B) = (x + (A + B)))
5	3, 4	mp3an3 903	. . . . . . . . 9 ⊢ ((x ∈ ℂ ⋀ A ∈ ℂ) → ((x + A) + B) = (x + (A + B)))
6		addcan.3	. . . . . . . . . 10 ⊢ C ∈ ℂ
7		axaddass 5257	. . . . . . . . . 10 ⊢ ((x ∈ ℂ ⋀ A ∈ ℂ ⋀ C ∈ ℂ) → ((x + A) + C) = (x + (A + C)))
8	6, 7	mp3an3 903	. . . . . . . . 9 ⊢ ((x ∈ ℂ ⋀ A ∈ ℂ) → ((x + A) + C) = (x + (A + C)))
9	5, 8	eqeq12d 1486	. . . . . . . 8 ⊢ ((x ∈ ℂ ⋀ A ∈ ℂ) → (((x + A) + B) = ((x + A) + C) ↔ (x + (A + B)) = (x + (A + C))))
10	1, 9	mpan2 695	. . . . . . 7 ⊢ (x ∈ ℂ → (((x + A) + B) = ((x + A) + C) ↔ (x + (A + B)) = (x + (A + C))))
11		opreq2 3960	. . . . . . 7 ⊢ ((A + B) = (A + C) → (x + (A + B)) = (x + (A + C)))
12	10, 11	syl5bir 210	. . . . . 6 ⊢ (x ∈ ℂ → ((A + B) = (A + C) → ((x + A) + B) = ((x + A) + C)))
13	12	adantr 389	. . . . 5 ⊢ ((x ∈ ℂ ⋀ (A + x) = 0) → ((A + B) = (A + C) → ((x + A) + B) = ((x + A) + C)))
14		axaddcom 5255	. . . . . . . . 9 ⊢ ((A ∈ ℂ ⋀ x ∈ ℂ) → (A + x) = (x + A))
15	1, 14	mpan 694	. . . . . . . 8 ⊢ (x ∈ ℂ → (A + x) = (x + A))
16	15	eqeq1d 1480	. . . . . . 7 ⊢ (x ∈ ℂ → ((A + x) = 0 ↔ (x + A) = 0))
17		opreq1 3959	. . . . . . . . 9 ⊢ ((x + A) = 0 → ((x + A) + B) = (0 + B))
18	3	addid2 5311	. . . . . . . . 9 ⊢ (0 + B) = B
19	17, 18	syl6eq 1520	. . . . . . . 8 ⊢ ((x + A) = 0 → ((x + A) + B) = B)
20		opreq1 3959	. . . . . . . . 9 ⊢ ((x + A) = 0 → ((x + A) + C) = (0 + C))
21	6	addid2 5311	. . . . . . . . 9 ⊢ (0 + C) = C
22	20, 21	syl6eq 1520	. . . . . . . 8 ⊢ ((x + A) = 0 → ((x + A) + C) = C)
23	19, 22	eqeq12d 1486	. . . . . . 7 ⊢ ((x + A) = 0 → (((x + A) + B) = ((x + A) + C) ↔ B = C))
24	16, 23	syl6bi 214	. . . . . 6 ⊢ (x ∈ ℂ → ((A + x) = 0 → (((x + A) + B) = ((x + A) + C) ↔ B = C)))
25	24	imp 350	. . . . 5 ⊢ ((x ∈ ℂ ⋀ (A + x) = 0) → (((x + A) + B) = ((x + A) + C) ↔ B = C))
26	13, 25	sylibd 202	. . . 4 ⊢ ((x ∈ ℂ ⋀ (A + x) = 0) → ((A + B) = (A + C) → B = C))
27	26	r19.23aiva 1741	. . 3 ⊢ (∃x ∈ ℂ (A + x) = 0 → ((A + B) = (A + C) → B = C))
28	2, 27	ax-mp 7	. 2 ⊢ ((A + B) = (A + C) → B = C)
29		opreq2 3960	. 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 954   ∈ wcel 956  ∃wrex 1643  (class class class)co 3954  ℂcc 5212  0cc0 5214   + caddc 5217

This theorem is referenced by:  addcant 5332  nn0opth 6604  cru 6675  cjreb 6724  pjnel 9608  lnopunilem1 9873

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226

Copyright terms: Public domain