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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 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
| 3 | 2 | renegcld 11691 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -𝑥 ∈ ℝ) | 
| 4 | simpr 484 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
| 5 | 4 | renegcld 11691 | . . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) | 
| 6 | recn 11246 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
| 7 | recn 11246 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
| 8 | negcon2 11563 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) | |
| 9 | 6, 7, 8 | syl2an 596 | . . . . . . 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 7687 | . . . . 5 ⊢ (⊤ → (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦))) | 
| 12 | 11 | mptru 1546 | . . . 4 ⊢ (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)) | 
| 13 | 12 | simpli 483 | . . 3 ⊢ 𝐹:ℝ–1-1-onto→ℝ | 
| 14 | ltneg 11764 | . . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑦 < -𝑧)) | |
| 15 | negex 11507 | . . . . . . 7 ⊢ -𝑧 ∈ V | |
| 16 | negex 11507 | . . . . . . 7 ⊢ -𝑦 ∈ V | |
| 17 | 15, 16 | brcnv 5892 | . . . . . 6 ⊢ (-𝑧◡ < -𝑦 ↔ -𝑦 < -𝑧) | 
| 18 | 14, 17 | bitr4di 289 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑧◡ < -𝑦)) | 
| 19 | negeq 11501 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → -𝑥 = -𝑧) | |
| 20 | 19, 1, 15 | fvmpt 7015 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (𝐹‘𝑧) = -𝑧) | 
| 21 | negeq 11501 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → -𝑥 = -𝑦) | |
| 22 | 21, 1, 16 | fvmpt 7015 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = -𝑦) | 
| 23 | 20, 22 | breqan12d 5158 | . . . . 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 3198 | . . 3 ⊢ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦)) | 
| 26 | df-isom 6569 | . . 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 711 | . 2 ⊢ 𝐹 Isom < , ◡ < (ℝ, ℝ) | 
| 28 | negeq 11501 | . . . 4 ⊢ (𝑦 = 𝑥 → -𝑦 = -𝑥) | |
| 29 | 28 | cbvmptv 5254 | . . 3 ⊢ (𝑦 ∈ ℝ ↦ -𝑦) = (𝑥 ∈ ℝ ↦ -𝑥) | 
| 30 | 12 | simpri 485 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦) | 
| 31 | 29, 30, 1 | 3eqtr4i 2774 | . 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 206 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 ↦ cmpt 5224 ◡ccnv 5683 –1-1-onto→wf1o 6559 ‘cfv 6560 Isom wiso 6561 ℂcc 11154 ℝcr 11155 < clt 11296 -cneg 11494 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 | 
| This theorem is referenced by: infrenegsup 12252 | 
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