![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
3 | 2 | renegcld 11638 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -𝑥 ∈ ℝ) |
4 | simpr 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
5 | 4 | renegcld 11638 | . . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) |
6 | recn 11196 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
7 | recn 11196 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | negcon2 11510 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) | |
9 | 6, 7, 8 | syl2an 595 | . . . . . . 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 7652 | . . . . 5 ⊢ (⊤ → (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦))) |
12 | 11 | mptru 1540 | . . . 4 ⊢ (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)) |
13 | 12 | simpli 483 | . . 3 ⊢ 𝐹:ℝ–1-1-onto→ℝ |
14 | ltneg 11711 | . . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑦 < -𝑧)) | |
15 | negex 11455 | . . . . . . 7 ⊢ -𝑧 ∈ V | |
16 | negex 11455 | . . . . . . 7 ⊢ -𝑦 ∈ V | |
17 | 15, 16 | brcnv 5872 | . . . . . 6 ⊢ (-𝑧◡ < -𝑦 ↔ -𝑦 < -𝑧) |
18 | 14, 17 | bitr4di 289 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑧◡ < -𝑦)) |
19 | negeq 11449 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → -𝑥 = -𝑧) | |
20 | 19, 1, 15 | fvmpt 6988 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (𝐹‘𝑧) = -𝑧) |
21 | negeq 11449 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → -𝑥 = -𝑦) | |
22 | 21, 1, 16 | fvmpt 6988 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = -𝑦) |
23 | 20, 22 | breqan12d 5154 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑧)◡ < (𝐹‘𝑦) ↔ -𝑧◡ < -𝑦)) |
24 | 18, 23 | bitr4d 282 | . . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦))) |
25 | 24 | rgen2 3189 | . . 3 ⊢ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦)) |
26 | df-isom 6542 | . . 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 708 | . 2 ⊢ 𝐹 Isom < , ◡ < (ℝ, ℝ) |
28 | negeq 11449 | . . . 4 ⊢ (𝑦 = 𝑥 → -𝑦 = -𝑥) | |
29 | 28 | cbvmptv 5251 | . . 3 ⊢ (𝑦 ∈ ℝ ↦ -𝑦) = (𝑥 ∈ ℝ ↦ -𝑥) |
30 | 12 | simpri 485 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦) |
31 | 29, 30, 1 | 3eqtr4i 2762 | . 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 205 ∧ wa 395 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∀wral 3053 class class class wbr 5138 ↦ cmpt 5221 ◡ccnv 5665 –1-1-onto→wf1o 6532 ‘cfv 6533 Isom wiso 6534 ℂcc 11104 ℝcr 11105 < clt 11245 -cneg 11442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 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 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 |
This theorem is referenced by: infrenegsup 12194 |
Copyright terms: Public domain | W3C validator |