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Mirrors > Home > MPE Home > Th. List > addsqn2reu | Structured version Visualization version GIF version |
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 26016, 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.) |
Ref | Expression |
---|---|
addsqn2reu | ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10595 | . . 3 ⊢ 1 ∈ ℂ | |
2 | neg1cn 11752 | . . 3 ⊢ -1 ∈ ℂ | |
3 | 1nn 11649 | . . . 4 ⊢ 1 ∈ ℕ | |
4 | nnneneg 11673 | . . . 4 ⊢ (1 ∈ ℕ → 1 ≠ -1) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1 ≠ -1 |
6 | 1, 2, 5 | 3pm3.2i 1335 | . 2 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) |
7 | 1cnd 10636 | . . . . 5 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ) | |
8 | negeu 10876 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶) | |
9 | 7, 8 | mpancom 686 | . . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶) |
10 | sq1 13559 | . . . . . . . . 9 ⊢ (1↑2) = 1 | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1↑2) = 1) |
12 | 11 | oveq2d 7172 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (𝑎 + 1)) |
13 | id 22 | . . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 𝑎 ∈ ℂ) | |
14 | 1cnd 10636 | . . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 1 ∈ ℂ) | |
15 | 13, 14 | addcomd 10842 | . . . . . . . 8 ⊢ (𝑎 ∈ ℂ → (𝑎 + 1) = (1 + 𝑎)) |
16 | 15 | adantl 484 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + 1) = (1 + 𝑎)) |
17 | 12, 16 | eqtrd 2856 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (1 + 𝑎)) |
18 | 17 | eqeq1d 2823 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶)) |
19 | 18 | reubidva 3388 | . . . 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 13560 | . . . . . . . . 9 ⊢ (-1↑2) = 1 | |
22 | 21 | a1i 11 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (-1↑2) = 1) |
23 | 22 | oveq2d 7172 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (𝑎 + 1)) |
24 | 23, 16 | eqtrd 2856 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (1 + 𝑎)) |
25 | 24 | eqeq1d 2823 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (-1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶)) |
26 | 25 | reubidva 3388 | . . . 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 514 | . 2 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)) |
29 | oveq1 7163 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2)) | |
30 | 29 | oveq2d 7172 | . . . . 5 ⊢ (𝑏 = 1 → (𝑎 + (𝑏↑2)) = (𝑎 + (1↑2))) |
31 | 30 | eqeq1d 2823 | . . . 4 ⊢ (𝑏 = 1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (1↑2)) = 𝐶)) |
32 | 31 | reubidv 3389 | . . 3 ⊢ (𝑏 = 1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶)) |
33 | oveq1 7163 | . . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2)) | |
34 | 33 | oveq2d 7172 | . . . . 5 ⊢ (𝑏 = -1 → (𝑎 + (𝑏↑2)) = (𝑎 + (-1↑2))) |
35 | 34 | eqeq1d 2823 | . . . 4 ⊢ (𝑏 = -1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (-1↑2)) = 𝐶)) |
36 | 35 | reubidv 3389 | . . 3 ⊢ (𝑏 = -1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)) |
37 | 32, 36 | 2nreu 4393 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃!wreu 3140 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 -cneg 10871 ℕcn 11638 2c2 11693 ↑cexp 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: (None) |
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