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Mirrors > Home > MPE Home > Th. List > negiso | Structured version Visualization version GIF version |
Description: Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
negiso.1 | ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥) |
Ref | Expression |
---|---|
negiso | ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negiso.1 | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥) | |
2 | simpr 476 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
3 | 2 | renegcld 10336 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -𝑥 ∈ ℝ) |
4 | simpr 476 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
5 | 4 | renegcld 10336 | . . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) |
6 | recn 9905 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
7 | recn 9905 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | negcon2 10213 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) | |
9 | 6, 7, 8 | syl2an 493 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
10 | 9 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
11 | 1, 3, 5, 10 | f1ocnv2d 6784 | . . . . 5 ⊢ (⊤ → (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦))) |
12 | 11 | trud 1484 | . . . 4 ⊢ (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)) |
13 | 12 | simpli 473 | . . 3 ⊢ 𝐹:ℝ–1-1-onto→ℝ |
14 | ltneg 10407 | . . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑦 < -𝑧)) | |
15 | negex 10158 | . . . . . . 7 ⊢ -𝑧 ∈ V | |
16 | negex 10158 | . . . . . . 7 ⊢ -𝑦 ∈ V | |
17 | 15, 16 | brcnv 5227 | . . . . . 6 ⊢ (-𝑧◡ < -𝑦 ↔ -𝑦 < -𝑧) |
18 | 14, 17 | syl6bbr 277 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑧◡ < -𝑦)) |
19 | negeq 10152 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → -𝑥 = -𝑧) | |
20 | 19, 1, 15 | fvmpt 6191 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (𝐹‘𝑧) = -𝑧) |
21 | negeq 10152 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → -𝑥 = -𝑦) | |
22 | 21, 1, 16 | fvmpt 6191 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = -𝑦) |
23 | 20, 22 | breqan12d 4599 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑧)◡ < (𝐹‘𝑦) ↔ -𝑧◡ < -𝑦)) |
24 | 18, 23 | bitr4d 270 | . . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦))) |
25 | 24 | rgen2a 2960 | . . 3 ⊢ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦)) |
26 | df-isom 5813 | . . 3 ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ↔ (𝐹:ℝ–1-1-onto→ℝ ∧ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦)))) | |
27 | 13, 25, 26 | mpbir2an 957 | . 2 ⊢ 𝐹 Isom < , ◡ < (ℝ, ℝ) |
28 | negeq 10152 | . . . 4 ⊢ (𝑦 = 𝑥 → -𝑦 = -𝑥) | |
29 | 28 | cbvmptv 4678 | . . 3 ⊢ (𝑦 ∈ ℝ ↦ -𝑦) = (𝑥 ∈ ℝ ↦ -𝑥) |
30 | 12 | simpri 477 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦) |
31 | 29, 30, 1 | 3eqtr4i 2642 | . 2 ⊢ ◡𝐹 = 𝐹 |
32 | 27, 31 | pm3.2i 470 | 1 ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ↦ cmpt 4643 ◡ccnv 5037 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 ℂcc 9813 ℝcr 9814 < clt 9953 -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-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-isom 5813 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-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 |
This theorem is referenced by: infrenegsup 10883 |
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