Theorem negeu 5327

Description: Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18.

Hypotheses

Ref	Expression
negeu.1	⊢ A ∈ ℂ
negeu.2	⊢ B ∈ ℂ

Assertion

Ref	Expression
negeu	⊢ ∃!x ∈ ℂ (A + x) = B

Distinct variable groups:   x,A   x,B

Proof of Theorem negeu

Step	Hyp	Ref	Expression
1		opreq2 3954	. . . 4 ⊢ (x = y → (A + x) = (A + y))
2	1	eqeq1d 1475	. . 3 ⊢ (x = y → ((A + x) = B ↔ (A + y) = B))
3	2	reu4 1924	. 2 ⊢ (∃!x ∈ ℂ (A + x) = B ↔ (∃x ∈ ℂ (A + x) = B ⋀ ∀x ∈ ℂ ∀y ∈ ℂ (((A + x) = B ⋀ (A + y) = B) → x = y)))
4		negeu.1	. . . 4 ⊢ A ∈ ℂ
5	4	cnegex 5321	. . 3 ⊢ ∃y ∈ ℂ (A + y) = 0
6		negeu.2	. . . . . . 7 ⊢ B ∈ ℂ
7		axaddcl 5243	. . . . . . 7 ⊢ ((y ∈ ℂ ⋀ B ∈ ℂ) → (y + B) ∈ ℂ)
8	6, 7	mpan2 694	. . . . . 6 ⊢ (y ∈ ℂ → (y + B) ∈ ℂ)
9		risset 1677	. . . . . 6 ⊢ ((y + B) ∈ ℂ ↔ ∃x ∈ ℂ x = (y + B))
10	8, 9	sylib 198	. . . . 5 ⊢ (y ∈ ℂ → ∃x ∈ ℂ x = (y + B))
11		opreq2 3954	. . . . . . . . . . . . 13 ⊢ (x = (y + B) → (A + x) = (A + (y + B)))
12		axaddass 5249	. . . . . . . . . . . . . . 15 ⊢ ((A ∈ ℂ ⋀ y ∈ ℂ ⋀ B ∈ ℂ) → ((A + y) + B) = (A + (y + B)))
13	4, 6, 12	mp3an13 904	. . . . . . . . . . . . . 14 ⊢ (y ∈ ℂ → ((A + y) + B) = (A + (y + B)))
14	13	eqcomd 1472	. . . . . . . . . . . . 13 ⊢ (y ∈ ℂ → (A + (y + B)) = ((A + y) + B))
15	11, 14	sylan9eqr 1521	. . . . . . . . . . . 12 ⊢ ((y ∈ ℂ ⋀ x = (y + B)) → (A + x) = ((A + y) + B))
16		opreq1 3953	. . . . . . . . . . . . 13 ⊢ ((A + y) = 0 → ((A + y) + B) = (0 + B))
17	6	addid2 5303	. . . . . . . . . . . . 13 ⊢ (0 + B) = B
18	16, 17	syl6eq 1515	. . . . . . . . . . . 12 ⊢ ((A + y) = 0 → ((A + y) + B) = B)
19	15, 18	sylan9eqr 1521	. . . . . . . . . . 11 ⊢ (((A + y) = 0 ⋀ (y ∈ ℂ ⋀ x = (y + B))) → (A + x) = B)
20	19	exp32 377	. . . . . . . . . 10 ⊢ ((A + y) = 0 → (y ∈ ℂ → (x = (y + B) → (A + x) = B)))
21	20	impcom 351	. . . . . . . . 9 ⊢ ((y ∈ ℂ ⋀ (A + y) = 0) → (x = (y + B) → (A + x) = B))
22	21	a1d 12	. . . . . . . 8 ⊢ ((y ∈ ℂ ⋀ (A + y) = 0) → (x ∈ ℂ → (x = (y + B) → (A + x) = B)))
23	22	r19.21aiv 1705	. . . . . . 7 ⊢ ((y ∈ ℂ ⋀ (A + y) = 0) → ∀x ∈ ℂ (x = (y + B) → (A + x) = B))
24	23	ex 373	. . . . . 6 ⊢ (y ∈ ℂ → ((A + y) = 0 → ∀x ∈ ℂ (x = (y + B) → (A + x) = B)))
25		r19.22 1723	. . . . . 6 ⊢ (∀x ∈ ℂ (x = (y + B) → (A + x) = B) → (∃x ∈ ℂ x = (y + B) → ∃x ∈ ℂ (A + x) = B))
26	24, 25	syl6 22	. . . . 5 ⊢ (y ∈ ℂ → ((A + y) = 0 → (∃x ∈ ℂ x = (y + B) → ∃x ∈ ℂ (A + x) = B)))
27	10, 26	mpid 47	. . . 4 ⊢ (y ∈ ℂ → ((A + y) = 0 → ∃x ∈ ℂ (A + x) = B))
28	27	r19.23aiv 1735	. . 3 ⊢ (∃y ∈ ℂ (A + y) = 0 → ∃x ∈ ℂ (A + x) = B)
29	5, 28	ax-mp 7	. 2 ⊢ ∃x ∈ ℂ (A + x) = B
30		addcant 5324	. . . . 5 ⊢ ((A ∈ ℂ ⋀ x ∈ ℂ ⋀ y ∈ ℂ) → ((A + x) = (A + y) ↔ x = y))
31		eqtr3t 1486	. . . . 5 ⊢ (((A + x) = B ⋀ (A + y) = B) → (A + x) = (A + y))
32	30, 31	syl5bi 208	. . . 4 ⊢ ((A ∈ ℂ ⋀ x ∈ ℂ ⋀ y ∈ ℂ) → (((A + x) = B ⋀ (A + y) = B) → x = y))
33	4, 32	mp3an1 900	. . 3 ⊢ ((x ∈ ℂ ⋀ y ∈ ℂ) → (((A + x) = B ⋀ (A + y) = B) → x = y))
34	33	rgen2a 1691	. 2 ⊢ ∀x ∈ ℂ ∀y ∈ ℂ (((A + x) = B ⋀ (A + y) = B) → x = y)
35	3, 29, 34	mpbir2an 728	1 ⊢ ∃!x ∈ ℂ (A + x) = B

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 773   = wceq 953   ∈ wcel 955  ∀wral 1637  ∃wrex 1638  ∃!wreu 1639  (class class class)co 3948  ℂcc 5204  0cc0 5206   + caddc 5209

This theorem is referenced by:  subcl 5338  subadd 5343

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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