Theorem addsqn2reu 27685

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 27684, 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 11186	. . 3 ⊢ 1 ∈ ℂ
2		neg1cn 12231	. . 3 ⊢ -1 ∈ ℂ
3		1nn 12272	. . . 4 ⊢ 1 ∈ ℕ
4		nnneneg 12299	. . . 4 ⊢ (1 ∈ ℕ → 1 ≠ -1)
5	3, 4	ax-mp 5	. . 3 ⊢ 1 ≠ -1
6	1, 2, 5	3pm3.2i 1358	. 2 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1)
7		1cnd 11230	. . . . 5 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
8		negeu 11475	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
9	7, 8	mpancom 701	. . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
10		sq1 14263	. . . . . . . . 9 ⊢ (1↑2) = 1
11	10	a1i 11	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1↑2) = 1)
12	11	oveq2d 7433	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (𝑎 + 1))
13		id 23	. . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 𝑎 ∈ ℂ)
14		1cnd 11230	. . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 1 ∈ ℂ)
15	13, 14	addcomd 11440	. . . . . . . 8 ⊢ (𝑎 ∈ ℂ → (𝑎 + 1) = (1 + 𝑎))
16	15	adantl 487	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + 1) = (1 + 𝑎))
17	12, 16	eqtrd 2797	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (1 + 𝑎))
18	17	eqeq1d 2764	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
19	18	reubidva 3381	. . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
20	9, 19	mpbird 260	. . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶)
21		neg1sqe1 14264	. . . . . . . . 9 ⊢ (-1↑2) = 1
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (-1↑2) = 1)
23	22	oveq2d 7433	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (𝑎 + 1))
24	23, 16	eqtrd 2797	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (1 + 𝑎))
25	24	eqeq1d 2764	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (-1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
26	25	reubidva 3381	. . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
27	9, 26	mpbird 260	. . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)
28	20, 27	jca 521	. 2 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
29		oveq1 7424	. . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2))
30	29	oveq2d 7433	. . . . 5 ⊢ (𝑏 = 1 → (𝑎 + (𝑏↑2)) = (𝑎 + (1↑2)))
31	30	eqeq1d 2764	. . . 4 ⊢ (𝑏 = 1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (1↑2)) = 𝐶))
32	31	reubidv 3383	. . 3 ⊢ (𝑏 = 1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶))
33		oveq1 7424	. . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2))
34	33	oveq2d 7433	. . . . 5 ⊢ (𝑏 = -1 → (𝑎 + (𝑏↑2)) = (𝑎 + (-1↑2)))
35	34	eqeq1d 2764	. . . 4 ⊢ (𝑏 = -1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (-1↑2)) = 𝐶))
36	35	reubidv 3383	. . 3 ⊢ (𝑏 = -1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
37	32, 36	2nreu 4405	. 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 69	1 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃!wreu 3365  (class class class)co 7417  ℂcc 11126  1c1 11129   + caddc 11131  -cneg 11470  ℕcn 12261  2c2 12323  ↑cexp 14129

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-n0 12533  df-z 12620  df-uz 12892  df-seq 14070  df-exp 14130

This theorem is used by: (None)