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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 488 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
3 | 2 | renegcld 11056 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -𝑥 ∈ ℝ) |
4 | simpr 488 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
5 | 4 | renegcld 11056 | . . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) |
6 | recn 10616 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
7 | recn 10616 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | negcon2 10928 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) | |
9 | 6, 7, 8 | syl2an 598 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
10 | 9 | adantl 485 | . . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
11 | 1, 3, 5, 10 | f1ocnv2d 7378 | . . . . 5 ⊢ (⊤ → (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦))) |
12 | 11 | mptru 1545 | . . . 4 ⊢ (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)) |
13 | 12 | simpli 487 | . . 3 ⊢ 𝐹:ℝ–1-1-onto→ℝ |
14 | ltneg 11129 | . . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑦 < -𝑧)) | |
15 | negex 10873 | . . . . . . 7 ⊢ -𝑧 ∈ V | |
16 | negex 10873 | . . . . . . 7 ⊢ -𝑦 ∈ V | |
17 | 15, 16 | brcnv 5717 | . . . . . 6 ⊢ (-𝑧◡ < -𝑦 ↔ -𝑦 < -𝑧) |
18 | 14, 17 | syl6bbr 292 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑧◡ < -𝑦)) |
19 | negeq 10867 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → -𝑥 = -𝑧) | |
20 | 19, 1, 15 | fvmpt 6745 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (𝐹‘𝑧) = -𝑧) |
21 | negeq 10867 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → -𝑥 = -𝑦) | |
22 | 21, 1, 16 | fvmpt 6745 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = -𝑦) |
23 | 20, 22 | breqan12d 5046 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑧)◡ < (𝐹‘𝑦) ↔ -𝑧◡ < -𝑦)) |
24 | 18, 23 | bitr4d 285 | . . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦))) |
25 | 24 | rgen2 3168 | . . 3 ⊢ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦)) |
26 | df-isom 6333 | . . 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 710 | . 2 ⊢ 𝐹 Isom < , ◡ < (ℝ, ℝ) |
28 | negeq 10867 | . . . 4 ⊢ (𝑦 = 𝑥 → -𝑦 = -𝑥) | |
29 | 28 | cbvmptv 5133 | . . 3 ⊢ (𝑦 ∈ ℝ ↦ -𝑦) = (𝑥 ∈ ℝ ↦ -𝑥) |
30 | 12 | simpri 489 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦) |
31 | 29, 30, 1 | 3eqtr4i 2831 | . 2 ⊢ ◡𝐹 = 𝐹 |
32 | 27, 31 | pm3.2i 474 | 1 ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 ↦ cmpt 5110 ◡ccnv 5518 –1-1-onto→wf1o 6323 ‘cfv 6324 Isom wiso 6325 ℂcc 10524 ℝcr 10525 < clt 10664 -cneg 10860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 |
This theorem is referenced by: infrenegsup 11611 |
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