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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 27347, 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 27347). For more details see comment for addsqnreup 27350. (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 11542 | . 2 ⊢ (𝐶 ∈ ℂ → (𝐶 − 1) ∈ ℂ) | |
2 | oveq1 7421 | . . . . . 6 ⊢ (𝑎 = (𝐶 − 1) → (𝑎 + (𝑏↑2)) = ((𝐶 − 1) + (𝑏↑2))) | |
3 | 2 | eqeq1d 2729 | . . . . 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 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 11182 | . . . . 5 ⊢ 1 ∈ ℂ | |
8 | neg1cn 12342 | . . . . 5 ⊢ -1 ∈ ℂ | |
9 | 1nn 12239 | . . . . . 6 ⊢ 1 ∈ ℕ | |
10 | nnneneg 12263 | . . . . . 6 ⊢ (1 ∈ ℕ → 1 ≠ -1) | |
11 | 9, 10 | ax-mp 5 | . . . . 5 ⊢ 1 ≠ -1 |
12 | 7, 8, 11 | 3pm3.2i 1337 | . . . 4 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) |
13 | sq1 14176 | . . . . . 6 ⊢ (1↑2) = 1 | |
14 | 13 | eqcomi 2736 | . . . . 5 ⊢ 1 = (1↑2) |
15 | neg1sqe1 14177 | . . . . . 6 ⊢ (-1↑2) = 1 | |
16 | 15 | eqcomi 2736 | . . . . 5 ⊢ 1 = (-1↑2) |
17 | 14, 16 | pm3.2i 470 | . . . 4 ⊢ (1 = (1↑2) ∧ 1 = (-1↑2)) |
18 | oveq1 7421 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2)) | |
19 | 18 | eqeq2d 2738 | . . . . 5 ⊢ (𝑏 = 1 → (1 = (𝑏↑2) ↔ 1 = (1↑2))) |
20 | oveq1 7421 | . . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2)) | |
21 | 20 | eqeq2d 2738 | . . . . 5 ⊢ (𝑏 = -1 → (1 = (𝑏↑2) ↔ 1 = (-1↑2))) |
22 | 19, 21 | 2nreu 4437 | . . . 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 14100 | . . . . . . 7 ⊢ (𝑏 ∈ ℂ → (𝑏↑2) ∈ ℂ) | |
27 | 26 | adantl 481 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑏↑2) ∈ ℂ) |
28 | 24, 25, 27 | subaddd 11605 | . . . . 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 11225 | . . . . . . . 8 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ) | |
31 | 29, 30 | nncand 11592 | . . . . . . 7 ⊢ (𝐶 ∈ ℂ → (𝐶 − (𝐶 − 1)) = 1) |
32 | 31 | adantr 480 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝐶 − (𝐶 − 1)) = 1) |
33 | 32 | eqeq1d 2729 | . . . . 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 3387 | . . 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 3609 | 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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∃wrex 3065 ∃!wreu 3369 (class class class)co 7414 ℂcc 11122 1c1 11125 + caddc 11127 − cmin 11460 -cneg 11461 ℕcn 12228 2c2 12283 ↑cexp 14044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-seq 13985 df-exp 14045 |
This theorem is referenced by: (None) |
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