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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 26804, 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 26804). For more details see comment for addsqnreup 26807. (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 11474 | . 2 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ) | |
2 | oveq1 7369 | . . . . . 6 ⊢ (𝑎 = (𝐶 − 1) → (𝑎 + (𝑏↑2)) = ((𝐶 − 1) + (𝑏↑2))) | |
3 | 2 | eqeq1d 2739 | . . . . 5 ⊢ (𝑎 = (𝐶 − 1) → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
4 | 3 | reubidv 3374 | . . . 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 483 | . 2 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 = (𝐶 − 1)) → (¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ¬ ∃!𝑏 ∈ ℂ ((𝐶 − 1) + (𝑏↑2)) = 𝐶)) |
7 | ax-1cn 11116 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | neg1cn 12274 | . . . . 5 ⊢ -1 ∈ ℂ | |
9 | 1nn 12171 | . . . . . 6 ⊢ 1 ∈ ℕ | |
10 | nnneneg 12195 | . . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ 1 ≠ -1 |
12 | 7, 8, 11 | 3pm3.2i 1340 | . . . 4 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) |
13 | sq1 14106 | . . . . . 6 ⊢ (1↑2) = 1 | |
14 | 13 | eqcomi 2746 | . . . . 5 ⊢ 1 = (1↑2) |
15 | neg1sqe1 14107 | . . . . . 6 ⊢ (-1↑2) = 1 | |
16 | 15 | eqcomi 2746 | . . . . 5 ⊢ 1 = (-1↑2) |
17 | 14, 16 | pm3.2i 472 | . . . 4 ⊢ (1 = (1↑2) ∧ 1 = (-1↑2)) |
18 | oveq1 7369 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2)) | |
19 | 18 | eqeq2d 2748 | . . . . 5 ⊢ (𝑏 = 1 → (1 = (𝑏↑2) ↔ 1 = (1↑2))) |
20 | oveq1 7369 | . . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2)) | |
21 | 20 | eqeq2d 2748 | . . . . 5 ⊢ (𝑏 = -1 → (1 = (𝑏↑2) ↔ 1 = (-1↑2))) |
22 | 19, 21 | 2nreu 4406 | . . . 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 484 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → 𝐶 ∈ ℂ) | |
25 | 1 | adantr 482 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − 1) ∈ ℂ) |
26 | sqcl 14030 | . . . . . . 7 ⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ) | |
27 | 26 | adantl 483 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ) |
28 | 24, 25, 27 | subaddd 11537 | . . . . 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 11157 | . . . . . . . 8 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ) | |
31 | 29, 30 | nncand 11524 | . . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 − (𝐶 − 1)) = 1) |
32 | 31 | adantr 482 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − (𝐶 − 1)) = 1) |
33 | 32 | eqeq1d 2739 | . . . . 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 3372 | . . 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 3586 | 1 ⊢ (𝐶 ∈ ℂ → ∃𝑎 ∈ ℂ ¬ ∃!𝑏 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∃wrex 3074 ∃!wreu 3354 (class class class)co 7362 ℂcc 11056 1c1 11059 + caddc 11061 − cmin 11392 -cneg 11393 ℕcn 12160 2c2 12215 ↑cexp 13974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-n0 12421 df-z 12507 df-uz 12771 df-seq 13914 df-exp 13975 |
This theorem is referenced by: (None) |
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