Theorem addsq2reu 27419

Description: For each complex number 𝐶, there exists a unique complex number 𝑎 added to the square of a unique another complex number 𝑏 resulting in the given complex number 𝐶. The unique complex number 𝑎 is 𝐶, and the unique another complex number 𝑏 is 0.

Remark: This, together with addsqnreup 27422, is an example showing that the pattern ∃!𝑎 ∈ 𝐴∃!𝑏 ∈ 𝐵𝜑 does not necessarily mean "There are unique sets 𝑎 and 𝑏 fulfilling 𝜑). See also comments for df-eu 2570 and 2eu4 2656. For more details see comment for addsqnreup 27422. (Contributed by AV, 21-Jun-2023.)

Assertion

Ref	Expression
addsq2reu	⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Distinct variable group:   𝐶,𝑎,𝑏

Proof of Theorem addsq2reu

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝐶 ∈ ℂ → 𝐶 ∈ ℂ)
2		oveq1 7375	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 + (𝑏↑2)) = (𝐶 + (𝑏↑2)))
3	2	eqeq1d 2739	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝐶 + (𝑏↑2)) = 𝐶))
4	3	reubidv 3368	. . . . 5 ⊢ (𝑎 = 𝐶 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶))
5		eqeq1 2741	. . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 = 𝑐 ↔ 𝐶 = 𝑐))
6	5	imbi2d 340	. . . . . 6 ⊢ (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐) ↔ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))
7	6	ralbidv 3161	. . . . 5 ⊢ (𝑎 = 𝐶 → (∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))
8	4, 7	anbi12d 633	. . . 4 ⊢ (𝑎 = 𝐶 → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))))
9	8	adantl 481	. . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 = 𝐶) → ((∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)) ↔ (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))))
10		0cnd 11137	. . . . . 6 ⊢ (𝐶 ∈ ℂ → 0 ∈ ℂ)
11		reueq 3697	. . . . . 6 ⊢ (0 ∈ ℂ ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0)
12	10, 11	sylib 218	. . . . 5 ⊢ (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ 𝑏 = 0)
13		subid 11412	. . . . . . . . 9 ⊢ (𝐶 ∈ ℂ → (𝐶 − 𝐶) = 0)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 𝐶) = 0)
15	14	eqeq1d 2739	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ 0 = (𝑏↑2)))
16		simpl 482	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
17		simpr 484	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝑏 ∈ ℂ)
18	17	sqcld 14079	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
19	16, 16, 18	subaddd 11522	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − 𝐶) = (𝑏↑2) ↔ (𝐶 + (𝑏↑2)) = 𝐶))
20		eqcom 2744	. . . . . . . . 9 ⊢ (0 = (𝑏↑2) ↔ (𝑏↑2) = 0)
21		sqeq0 14055	. . . . . . . . 9 ⊢ (𝑏 ∈ ℂ → ((𝑏↑2) = 0 ↔ 𝑏 = 0))
22	20, 21	bitrid 283	. . . . . . . 8 ⊢ (𝑏 ∈ ℂ → (0 = (𝑏↑2) ↔ 𝑏 = 0))
23	22	adantl 481	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (0 = (𝑏↑2) ↔ 𝑏 = 0))
24	15, 19, 23	3bitr3d 309	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 + (𝑏↑2)) = 𝐶 ↔ 𝑏 = 0))
25	24	reubidva 3366	. . . . 5 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 𝑏 = 0))
26	12, 25	mpbird 257	. . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶)
27		simpr 484	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝑐 ∈ ℂ)
28	27	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝑐 ∈ ℂ)
29		sqcl 14053	. . . . . . . . 9 ⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ)
30	29	adantl 481	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ)
31		simpl 482	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → 𝐶 ∈ ℂ)
32	31	adantr 480	. . . . . . . 8 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ)
33	28, 30, 32	addrsub 11566	. . . . . . 7 ⊢ (((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) ∧ 𝑏 ∈ ℂ) → ((𝑐 + (𝑏↑2)) = 𝐶 ↔ (𝑏↑2) = (𝐶 − 𝑐)))
34	33	reubidva 3366	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐)))
35		subcl 11391	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (𝐶 − 𝑐) ∈ ℂ)
36		reusq0 15400	. . . . . . . 8 ⊢ ((𝐶 − 𝑐) ∈ ℂ → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) ↔ (𝐶 − 𝑐) = 0))
37	35, 36	syl 17	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) ↔ (𝐶 − 𝑐) = 0))
38		subeq0 11419	. . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 ↔ 𝐶 = 𝑐))
39	38	biimpd 229	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ((𝐶 − 𝑐) = 0 → 𝐶 = 𝑐))
40	37, 39	sylbid 240	. . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑏↑2) = (𝐶 − 𝑐) → 𝐶 = 𝑐))
41	34, 40	sylbid 240	. . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑐 ∈ ℂ) → (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))
42	41	ralrimiva 3130	. . . 4 ⊢ (𝐶 ∈ ℂ → ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐))
43	26, 42	jca 511	. . 3 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ (𝐶 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝐶 = 𝑐)))
44	1, 9, 43	rspcedvd 3580	. 2 ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)))
45		oveq1 7375	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 + (𝑏↑2)) = (𝑐 + (𝑏↑2)))
46	45	eqeq1d 2739	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑐 + (𝑏↑2)) = 𝐶))
47	46	reubidv 3368	. . 3 ⊢ (𝑎 = 𝑐 → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶))
48	47	reu8 3693	. 2 ⊢ (∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃𝑎 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ∧ ∀𝑐 ∈ ℂ (∃!𝑏 ∈ ℂ (𝑐 + (𝑏↑2)) = 𝐶 → 𝑎 = 𝑐)))
49	44, 48	sylibr 234	1 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350  (class class class)co 7368  ℂcc 11036  0cc0 11038   + caddc 11041   − cmin 11376  2c2 12212  ↑cexp 13996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-rp 12918  df-seq 13937  df-exp 13997  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171

This theorem is referenced by: (None)