Theorem addsqn2reu 27492

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 27491, 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 11124	. . 3 ⊢ 1 ∈ ℂ
2		neg1cn 12173	. . 3 ⊢ -1 ∈ ℂ
3		1nn 12214	. . . 4 ⊢ 1 ∈ ℕ
4		nnneneg 12241	. . . 4 ⊢ (1 ∈ ℕ → 1 ≠ -1)
5	3, 4	ax-mp 5	. . 3 ⊢ 1 ≠ -1
6	1, 2, 5	3pm3.2i 1352	. 2 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1)
7		1cnd 11168	. . . . 5 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ)
8		negeu 11413	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
9	7, 8	mpancom 698	. . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)
10		sq1 14201	. . . . . . . . 9 ⊢ (1↑2) = 1
11	10	a1i 11	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1↑2) = 1)
12	11	oveq2d 7406	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (𝑎 + 1))
13		id 22	. . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 𝑎 ∈ ℂ)
14		1cnd 11168	. . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 1 ∈ ℂ)
15	13, 14	addcomd 11378	. . . . . . . 8 ⊢ (𝑎 ∈ ℂ → (𝑎 + 1) = (1 + 𝑎))
16	15	adantl 485	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + 1) = (1 + 𝑎))
17	12, 16	eqtrd 2796	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (1 + 𝑎))
18	17	eqeq1d 2763	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
19	18	reubidva 3380	. . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
20	9, 19	mpbird 259	. . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶)
21		neg1sqe1 14202	. . . . . . . . 9 ⊢ (-1↑2) = 1
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (-1↑2) = 1)
23	22	oveq2d 7406	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (𝑎 + 1))
24	23, 16	eqtrd 2796	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (1 + 𝑎))
25	24	eqeq1d 2763	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (-1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶))
26	25	reubidva 3380	. . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶))
27	9, 26	mpbird 259	. . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)
28	20, 27	jca 519	. 2 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
29		oveq1 7397	. . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2))
30	29	oveq2d 7406	. . . . 5 ⊢ (𝑏 = 1 → (𝑎 + (𝑏↑2)) = (𝑎 + (1↑2)))
31	30	eqeq1d 2763	. . . 4 ⊢ (𝑏 = 1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (1↑2)) = 𝐶))
32	31	reubidv 3382	. . 3 ⊢ (𝑏 = 1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶))
33		oveq1 7397	. . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2))
34	33	oveq2d 7406	. . . . 5 ⊢ (𝑏 = -1 → (𝑎 + (𝑏↑2)) = (𝑎 + (-1↑2)))
35	34	eqeq1d 2763	. . . 4 ⊢ (𝑏 = -1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (-1↑2)) = 𝐶))
36	35	reubidv 3382	. . 3 ⊢ (𝑏 = -1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶))
37	32, 36	2nreu 4395	. 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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃!wreu 3364  (class class class)co 7390  ℂcc 11064  1c1 11067   + caddc 11069  -cneg 11408  ℕcn 12203  2c2 12265  ↑cexp 14067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  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 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-er 8671  df-en 8921  df-dom 8922  df-sdom 8923  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-div 11838  df-nn 12204  df-2 12273  df-n0 12475  df-z 12562  df-uz 12833  df-seq 14008  df-exp 14068

This theorem is referenced by: (None)