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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 25698, 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 10441 | . . 3 ⊢ 1 ∈ ℂ | |
2 | neg1cn 11599 | . . 3 ⊢ -1 ∈ ℂ | |
3 | 1nn 11497 | . . . 4 ⊢ 1 ∈ ℕ | |
4 | nnneneg 11520 | . . . 4 ⊢ (1 ∈ ℕ → 1 ≠ -1) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1 ≠ -1 |
6 | 1, 2, 5 | 3pm3.2i 1332 | . 2 ⊢ (1 ∈ ℂ ∧ -1 ∈ ℂ ∧ 1 ≠ -1) |
7 | 1cnd 10482 | . . . . 5 ⊢ (𝐶 ∈ ℂ → 1 ∈ ℂ) | |
8 | negeu 10723 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶) | |
9 | 7, 8 | mpancom 684 | . . . 4 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶) |
10 | sq1 13408 | . . . . . . . . 9 ⊢ (1↑2) = 1 | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (1↑2) = 1) |
12 | 11 | oveq2d 7032 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (𝑎 + 1)) |
13 | id 22 | . . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 𝑎 ∈ ℂ) | |
14 | 1cnd 10482 | . . . . . . . . 9 ⊢ (𝑎 ∈ ℂ → 1 ∈ ℂ) | |
15 | 13, 14 | addcomd 10689 | . . . . . . . 8 ⊢ (𝑎 ∈ ℂ → (𝑎 + 1) = (1 + 𝑎)) |
16 | 15 | adantl 482 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + 1) = (1 + 𝑎)) |
17 | 12, 16 | eqtrd 2831 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (1↑2)) = (1 + 𝑎)) |
18 | 17 | eqeq1d 2797 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶)) |
19 | 18 | reubidva 3347 | . . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)) |
20 | 9, 19 | mpbird 258 | . . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶) |
21 | neg1sqe1 13409 | . . . . . . . . 9 ⊢ (-1↑2) = 1 | |
22 | 21 | a1i 11 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (-1↑2) = 1) |
23 | 22 | oveq2d 7032 | . . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (𝑎 + 1)) |
24 | 23, 16 | eqtrd 2831 | . . . . . 6 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → (𝑎 + (-1↑2)) = (1 + 𝑎)) |
25 | 24 | eqeq1d 2797 | . . . . 5 ⊢ ((𝐶 ∈ ℂ ∧ 𝑎 ∈ ℂ) → ((𝑎 + (-1↑2)) = 𝐶 ↔ (1 + 𝑎) = 𝐶)) |
26 | 25 | reubidva 3347 | . . . 4 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (1 + 𝑎) = 𝐶)) |
27 | 9, 26 | mpbird 258 | . . 3 ⊢ (𝐶 ∈ ℂ → ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶) |
28 | 20, 27 | jca 512 | . 2 ⊢ (𝐶 ∈ ℂ → (∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶 ∧ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)) |
29 | oveq1 7023 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏↑2) = (1↑2)) | |
30 | 29 | oveq2d 7032 | . . . . 5 ⊢ (𝑏 = 1 → (𝑎 + (𝑏↑2)) = (𝑎 + (1↑2))) |
31 | 30 | eqeq1d 2797 | . . . 4 ⊢ (𝑏 = 1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (1↑2)) = 𝐶)) |
32 | 31 | reubidv 3349 | . . 3 ⊢ (𝑏 = 1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (1↑2)) = 𝐶)) |
33 | oveq1 7023 | . . . . . 6 ⊢ (𝑏 = -1 → (𝑏↑2) = (-1↑2)) | |
34 | 33 | oveq2d 7032 | . . . . 5 ⊢ (𝑏 = -1 → (𝑎 + (𝑏↑2)) = (𝑎 + (-1↑2))) |
35 | 34 | eqeq1d 2797 | . . . 4 ⊢ (𝑏 = -1 → ((𝑎 + (𝑏↑2)) = 𝐶 ↔ (𝑎 + (-1↑2)) = 𝐶)) |
36 | 35 | reubidv 3349 | . . 3 ⊢ (𝑏 = -1 → (∃!𝑎 ∈ ℂ (𝑎 + (𝑏↑2)) = 𝐶 ↔ ∃!𝑎 ∈ ℂ (𝑎 + (-1↑2)) = 𝐶)) |
37 | 32, 36 | 2nreu 4307 | . 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 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∃!wreu 3107 (class class class)co 7016 ℂcc 10381 1c1 10384 + caddc 10386 -cneg 10718 ℕcn 11486 2c2 11540 ↑cexp 13279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-seq 13220 df-exp 13280 |
This theorem is referenced by: (None) |
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