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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 1541 | . 2 ⊢ ⊤ | |
2 | nfriota1 7121 | . . . 4 ⊢ Ⅎ𝑦(℩𝑦 ∈ ℝ 𝜓) | |
3 | 2 | nfneg 10882 | . . 3 ⊢ Ⅎ𝑦-(℩𝑦 ∈ ℝ 𝜓) |
4 | renegcl 10949 | . . . 4 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ) | |
5 | 4 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) |
6 | simpr 487 | . . . 4 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) → (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) | |
7 | 6 | renegcld 11067 | . . 3 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) → -(℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) |
8 | riotaneg.1 | . . 3 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) | |
9 | negeq 10878 | . . 3 ⊢ (𝑦 = (℩𝑦 ∈ ℝ 𝜓) → -𝑦 = -(℩𝑦 ∈ ℝ 𝜓)) | |
10 | renegcl 10949 | . . . . 5 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
11 | recn 10627 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
12 | recn 10627 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
13 | negcon2 10939 | . . . . . 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 5321 | . . . 4 ⊢ (𝑥 ∈ ℝ → ∃!𝑦 ∈ ℝ 𝑥 = -𝑦) |
16 | 15 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → ∃!𝑦 ∈ ℝ 𝑥 = -𝑦) |
17 | 3, 5, 7, 8, 9, 16 | riotaxfrd 7148 | . 2 ⊢ ((⊤ ∧ ∃!𝑥 ∈ ℝ 𝜑) → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) |
18 | 1, 17 | mpan 688 | 1 ⊢ (∃!𝑥 ∈ ℝ 𝜑 → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ∃!wreu 3140 ℩crio 7113 ℂcc 10535 ℝcr 10536 -cneg 10871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 |
This theorem is referenced by: (None) |
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