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Mirrors > Home > MPE Home > Th. List > riotaneg | Structured version Visualization version GIF version |
Description: The negative of the unique real such that 𝜑. (Contributed by NM, 13-Jun-2005.) |
Ref | Expression |
---|---|
riotaneg.1 | ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
riotaneg | ⊢ (∃!𝑥 ∈ ℝ 𝜑 → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1546 | . 2 ⊢ ⊤ | |
2 | nfriota1 7367 | . . . 4 ⊢ Ⅎ𝑦(℩𝑦 ∈ ℝ 𝜓) | |
3 | 2 | nfneg 11452 | . . 3 ⊢ Ⅎ𝑦-(℩𝑦 ∈ ℝ 𝜓) |
4 | renegcl 11519 | . . . 4 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ) | |
5 | 4 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) |
6 | simpr 486 | . . . 4 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) → (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) | |
7 | 6 | renegcld 11637 | . . 3 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) → -(℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) |
8 | riotaneg.1 | . . 3 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) | |
9 | negeq 11448 | . . 3 ⊢ (𝑦 = (℩𝑦 ∈ ℝ 𝜓) → -𝑦 = -(℩𝑦 ∈ ℝ 𝜓)) | |
10 | renegcl 11519 | . . . . 5 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
11 | recn 11196 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
12 | recn 11196 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
13 | negcon2 11509 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) | |
14 | 11, 12, 13 | syl2an 597 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
15 | 10, 14 | reuhyp 5417 | . . . 4 ⊢ (𝑥 ∈ ℝ → ∃!𝑦 ∈ ℝ 𝑥 = -𝑦) |
16 | 15 | adantl 483 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → ∃!𝑦 ∈ ℝ 𝑥 = -𝑦) |
17 | 3, 5, 7, 8, 9, 16 | riotaxfrd 7395 | . 2 ⊢ ((⊤ ∧ ∃!𝑥 ∈ ℝ 𝜑) → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) |
18 | 1, 17 | mpan 689 | 1 ⊢ (∃!𝑥 ∈ ℝ 𝜑 → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ∃!wreu 3375 ℩crio 7359 ℂcc 11104 ℝcr 11105 -cneg 11441 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 |
This theorem is referenced by: (None) |
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