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Mirrors > Home > MPE Home > Th. List > addsqrexnreu | Structured version Visualization version GIF version |
Description: For each complex number,
there exists a complex number to which the
square of more than one (or no) other complex numbers can be added to
result in the given complex number.
Remark: This theorem, together with addsq2reu 26016, shows that there are cases in which there is a set together with a not unique other set fulfilling a wff, although there is a unique set fulfilling the wff together with another unique set (see addsq2reu 26016). For more details see comment for addsqnreup 26019. (Contributed by AV, 20-Jun-2023.) |
Ref | Expression |
---|---|
addsqrexnreu | ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ ¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2cnm 10952 | . 2 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ) | |
2 | oveq1 7163 | . . . . . 6 ⊢ (𝑎 = (𝐶 − 1) → (𝑎 + (𝑏↑2)) = ((𝐶 − 1) + (𝑏↑2))) | |
3 | 2 | eqeq1d 2823 | . . . . 5 ⊢ (𝑎 = (𝐶 − 1) → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
4 | 3 | reubidv 3389 | . . . 4 ⊢ (𝑎 = (𝐶 − 1) → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
5 | 4 | notbid 320 | . . 3 ⊢ (𝑎 = (𝐶 − 1) → (¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ¬ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
6 | 5 | adantl 484 | . 2 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 = (𝐶 − 1)) → (¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ¬ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
7 | ax-1cn 10595 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | neg1cn 11752 | . . . . 5 ⊢ -1 ∈ ℂ | |
9 | 1nn 11649 | . . . . . 6 ⊢ 1 ∈ ℕ | |
10 | nnneneg 11673 | . . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ 1 ≠ -1 |
12 | 7, 8, 11 | 3pm3.2i 1335 | . . . 4 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) |
13 | sq1 13559 | . . . . . 6 ⊢ (1↑2) = 1 | |
14 | 13 | eqcomi 2830 | . . . . 5 ⊢ 1 = (1↑2) |
15 | neg1sqe1 13560 | . . . . . 6 ⊢ (-1↑2) = 1 | |
16 | 15 | eqcomi 2830 | . . . . 5 ⊢ 1 = (-1↑2) |
17 | 14, 16 | pm3.2i 473 | . . . 4 ⊢ (1 = (1↑2) ∧ 1 = (-1↑2)) |
18 | oveq1 7163 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2)) | |
19 | 18 | eqeq2d 2832 | . . . . 5 ⊢ (𝑏 = 1 → (1 = (𝑏↑2) ↔ 1 = (1↑2))) |
20 | oveq1 7163 | . . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2)) | |
21 | 20 | eqeq2d 2832 | . . . . 5 ⊢ (𝑏 = -1 → (1 = (𝑏↑2) ↔ 1 = (-1↑2))) |
22 | 19, 21 | 2nreu 4393 | . . . 4 ⊢ ((1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) → ((1 = (1↑2) ∧ 1 = (-1↑2)) → ¬ ∃!𝑏 ∈ ℂ 1 = (𝑏↑2))) |
23 | 12, 17, 22 | mp2 9 | . . 3 ⊢ ¬ ∃!𝑏 ∈ ℂ 1 = (𝑏↑2) |
24 | simpl 485 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ) | |
25 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 1) ∈ ℂ) |
26 | sqcl 13485 | . . . . . . 7 ⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ) | |
27 | 26 | adantl 484 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ) |
28 | 24, 25, 27 | subaddd 11015 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − (𝐶 − 1)) = (𝑏↑2) ↔ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
29 | id 22 | . . . . . . . 8 ⊢ (𝐶 ∈ ℂ → 𝐶 ∈ ℂ) | |
30 | 1cnd 10636 | . . . . . . . 8 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ) | |
31 | 29, 30 | nncand 11002 | . . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 − (𝐶 − 1)) = 1) |
32 | 31 | adantr 483 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − (𝐶 − 1)) = 1) |
33 | 32 | eqeq1d 2823 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − (𝐶 − 1)) = (𝑏↑2) ↔ 1 = (𝑏↑2))) |
34 | 28, 33 | bitr3d 283 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (((𝐶 − 1) + (𝑏↑2)) = 𝐶 ↔ 1 = (𝑏↑2))) |
35 | 34 | reubidva 3388 | . . 3 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 1 = (𝑏↑2))) |
36 | 23, 35 | mtbiri 329 | . 2 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶) |
37 | 1, 6, 36 | rspcedvd 3626 | 1 ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ ¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∃!wreu 3140 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 − cmin 10870 -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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