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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 1479 | . 2 ⊢ ⊤ | |
2 | nfriota1 6518 | . . . 4 ⊢ Ⅎ𝑦(℩𝑦 ∈ ℝ 𝜓) | |
3 | 2 | nfneg 10156 | . . 3 ⊢ Ⅎ𝑦-(℩𝑦 ∈ ℝ 𝜓) |
4 | renegcl 10223 | . . . 4 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ) | |
5 | 4 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) |
6 | simpr 476 | . . . 4 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) → (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) | |
7 | 6 | renegcld 10336 | . . 3 ⊢ ((⊤ ∧ (℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) → -(℩𝑦 ∈ ℝ 𝜓) ∈ ℝ) |
8 | riotaneg.1 | . . 3 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜓)) | |
9 | negeq 10152 | . . 3 ⊢ (𝑦 = (℩𝑦 ∈ ℝ 𝜓) → -𝑦 = -(℩𝑦 ∈ ℝ 𝜓)) | |
10 | renegcl 10223 | . . . . 5 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) | |
11 | recn 9905 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
12 | recn 9905 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
13 | negcon2 10213 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) | |
14 | 11, 12, 13 | syl2an 493 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
15 | 10, 14 | reuhyp 4822 | . . . 4 ⊢ (𝑥 ∈ ℝ → ∃!𝑦 ∈ ℝ 𝑥 = -𝑦) |
16 | 15 | adantl 481 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → ∃!𝑦 ∈ ℝ 𝑥 = -𝑦) |
17 | 3, 5, 7, 8, 9, 16 | riotaxfrd 6541 | . 2 ⊢ ((⊤ ∧ ∃!𝑥 ∈ ℝ 𝜑) → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) |
18 | 1, 17 | mpan 702 | 1 ⊢ (∃!𝑥 ∈ ℝ 𝜑 → (℩𝑥 ∈ ℝ 𝜑) = -(℩𝑦 ∈ ℝ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∃!wreu 2898 ℩crio 6510 ℂcc 9813 ℝcr 9814 -cneg 10146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 |
This theorem is referenced by: (None) |
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