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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 27498, 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 27498). For more details see comment for addsqnreup 27501. (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 11572 | . 2 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ) | |
2 | oveq1 7437 | . . . . . 6 ⊢ (𝑎 = (𝐶 − 1) → (𝑎 + (𝑏↑2)) = ((𝐶 − 1) + (𝑏↑2))) | |
3 | 2 | eqeq1d 2736 | . . . . 5 ⊢ (𝑎 = (𝐶 − 1) → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
4 | 3 | reubidv 3395 | . . . 4 ⊢ (𝑎 = (𝐶 − 1) → (∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
5 | 4 | notbid 318 | . . 3 ⊢ (𝑎 = (𝐶 − 1) → (¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ¬ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
6 | 5 | adantl 481 | . 2 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 = (𝐶 − 1)) → (¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ¬ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
7 | ax-1cn 11210 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | neg1cn 12377 | . . . . 5 ⊢ -1 ∈ ℂ | |
9 | 1nn 12274 | . . . . . 6 ⊢ 1 ∈ ℕ | |
10 | nnneneg 12298 | . . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ 1 ≠ -1 |
12 | 7, 8, 11 | 3pm3.2i 1338 | . . . 4 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) |
13 | sq1 14230 | . . . . . 6 ⊢ (1↑2) = 1 | |
14 | 13 | eqcomi 2743 | . . . . 5 ⊢ 1 = (1↑2) |
15 | neg1sqe1 14231 | . . . . . 6 ⊢ (-1↑2) = 1 | |
16 | 15 | eqcomi 2743 | . . . . 5 ⊢ 1 = (-1↑2) |
17 | 14, 16 | pm3.2i 470 | . . . 4 ⊢ (1 = (1↑2) ∧ 1 = (-1↑2)) |
18 | oveq1 7437 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2)) | |
19 | 18 | eqeq2d 2745 | . . . . 5 ⊢ (𝑏 = 1 → (1 = (𝑏↑2) ↔ 1 = (1↑2))) |
20 | oveq1 7437 | . . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2)) | |
21 | 20 | eqeq2d 2745 | . . . . 5 ⊢ (𝑏 = -1 → (1 = (𝑏↑2) ↔ 1 = (-1↑2))) |
22 | 19, 21 | 2nreu 4449 | . . . 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 482 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ) | |
25 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 1) ∈ ℂ) |
26 | sqcl 14154 | . . . . . . 7 ⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ) | |
27 | 26 | adantl 481 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ) |
28 | 24, 25, 27 | subaddd 11635 | . . . . 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 11253 | . . . . . . . 8 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ) | |
31 | 29, 30 | nncand 11622 | . . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 − (𝐶 − 1)) = 1) |
32 | 31 | adantr 480 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − (𝐶 − 1)) = 1) |
33 | 32 | eqeq1d 2736 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → ((𝐶 − (𝐶 − 1)) = (𝑏↑2) ↔ 1 = (𝑏↑2))) |
34 | 28, 33 | bitr3d 281 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (((𝐶 − 1) + (𝑏↑2)) = 𝐶 ↔ 1 = (𝑏↑2))) |
35 | 34 | reubidva 3393 | . . 3 ⊢ (𝐶 ∈ ℂ → (∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶 ↔ ∃!𝑏 ∈ ℂ 1 = (𝑏↑2))) |
36 | 23, 35 | mtbiri 327 | . 2 ⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶) |
37 | 1, 6, 36 | rspcedvd 3623 | 1 ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ ¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 ∃!wreu 3375 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 − cmin 11489 -cneg 11490 ℕcn 12263 2c2 12318 ↑cexp 14098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-seq 14039 df-exp 14099 |
This theorem is referenced by: (None) |
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