Theorem addsqn2reu 26494

Description: For each complex number 𝐶, there does not exist a unique complex number 𝑏, squared and added to a unique another complex number 𝑎 resulting in the given complex number 𝐶. Actually, for each complex number 𝑏, 𝑎 = (𝐶 − (𝑏↑2)) is unique.

Remark: This, together with addsq2reu 26493, shows that commutation of two unique quantifications need not be equivalent, and provides an evident justification of the fact that considering the pair of variables is necessary to obtain what we intuitively understand as "double unique existence". (Proposed by GL, 23-Jun-2023.). (Contributed by AV, 23-Jun-2023.)

Assertion

Ref	Expression
addsqn2reu	⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Distinct variable group:   𝐶,𝑎,𝑏

Proof of Theorem addsqn2reu

Step	Hyp	Ref	Expression
1		ax-1cn 10860	. . 3 ⊢ 1 ∈ ℂ
2		neg1cn 12017	. . 3 ⊢ -1 ∈ ℂ
3		1nn 11914	. . . 4 ⊢ 1 ∈ ℕ
4		nnneneg 11938	. . . 4 ⊢ (1 ∈ ℕ → 1 ≠ -1)
5	3, 4	ax-mp 5	. . 3 ⊢ 1 ≠ -1
6	1, 2, 5	3pm3.2i 1337	. 2 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1)
7		1cnd 10901	. . . . 5 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
8		negeu 11141	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
9	7, 8	mpancom 684	. . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
10		sq1 13840	. . . . . . . . 9 ⊢ (1↑2) = 1
11	10	a1i 11	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1↑2) = 1)
12	11	oveq2d 7271	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (𝑎 + 1))
13		id 22	. . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 𝑎 ∈ ℂ)
14		1cnd 10901	. . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 1 ∈ ℂ)
15	13, 14	addcomd 11107	. . . . . . . 8 ⊢ (𝑎 ∈ ℂ → (𝑎 + 1) = (1 + 𝑎))
16	15	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + 1) = (1 + 𝑎))
17	12, 16	eqtrd 2778	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (1 + 𝑎))
18	17	eqeq1d 2740	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
19	18	reubidva 3314	. . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
20	9, 19	mpbird 256	. . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶)
21		neg1sqe1 13841	. . . . . . . . 9 ⊢ (-1↑2) = 1
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (-1↑2) = 1)
23	22	oveq2d 7271	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (𝑎 + 1))
24	23, 16	eqtrd 2778	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (1 + 𝑎))
25	24	eqeq1d 2740	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (-1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
26	25	reubidva 3314	. . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
27	9, 26	mpbird 256	. . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)
28	20, 27	jca 511	. 2 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
29		oveq1 7262	. . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2))
30	29	oveq2d 7271	. . . . 5 ⊢ (𝑏 = 1 → (𝑎 + (𝑏↑2)) = (𝑎 + (1↑2)))
31	30	eqeq1d 2740	. . . 4 ⊢ (𝑏 = 1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (1↑2)) = 𝐶))
32	31	reubidv 3315	. . 3 ⊢ (𝑏 = 1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶))
33		oveq1 7262	. . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2))
34	33	oveq2d 7271	. . . . 5 ⊢ (𝑏 = -1 → (𝑎 + (𝑏↑2)) = (𝑎 + (-1↑2)))
35	34	eqeq1d 2740	. . . 4 ⊢ (𝑏 = -1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (-1↑2)) = 𝐶))
36	35	reubidv 3315	. . 3 ⊢ (𝑏 = -1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
37	32, 36	2nreu 4372	. 2 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) → ((∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶) → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶))
38	6, 28, 37	mpsyl 68	1 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃!wreu 3065  (class class class)co 7255  ℂcc 10800  1c1 10803   + caddc 10805  -cneg 11136  ℕcn 11903  2c2 11958  ↑cexp 13710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-seq 13650  df-exp 13711

This theorem is referenced by: (None)