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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 485 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
3 | 2 | renegcld 11640 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -𝑥 ∈ ℝ) |
4 | simpr 485 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ) | |
5 | 4 | renegcld 11640 | . . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ) |
6 | recn 11199 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
7 | recn 11199 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ) | |
8 | negcon2 11512 | . . . . . . . 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 482 | . . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥)) |
11 | 1, 3, 5, 10 | f1ocnv2d 7658 | . . . . 5 ⊢ (⊤ → (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦))) |
12 | 11 | mptru 1548 | . . . 4 ⊢ (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)) |
13 | 12 | simpli 484 | . . 3 ⊢ 𝐹:ℝ–1-1-onto→ℝ |
14 | ltneg 11713 | . . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑦 < -𝑧)) | |
15 | negex 11457 | . . . . . . 7 ⊢ -𝑧 ∈ V | |
16 | negex 11457 | . . . . . . 7 ⊢ -𝑦 ∈ V | |
17 | 15, 16 | brcnv 5882 | . . . . . 6 ⊢ (-𝑧◡ < -𝑦 ↔ -𝑦 < -𝑧) |
18 | 14, 17 | bitr4di 288 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑧◡ < -𝑦)) |
19 | negeq 11451 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → -𝑥 = -𝑧) | |
20 | 19, 1, 15 | fvmpt 6998 | . . . . . 6 ⊢ (𝑧 ∈ ℝ → (𝐹‘𝑧) = -𝑧) |
21 | negeq 11451 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → -𝑥 = -𝑦) | |
22 | 21, 1, 16 | fvmpt 6998 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = -𝑦) |
23 | 20, 22 | breqan12d 5164 | . . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑧)◡ < (𝐹‘𝑦) ↔ -𝑧◡ < -𝑦)) |
24 | 18, 23 | bitr4d 281 | . . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦))) |
25 | 24 | rgen2 3197 | . . 3 ⊢ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦)) |
26 | df-isom 6552 | . . 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 709 | . 2 ⊢ 𝐹 Isom < , ◡ < (ℝ, ℝ) |
28 | negeq 11451 | . . . 4 ⊢ (𝑦 = 𝑥 → -𝑦 = -𝑥) | |
29 | 28 | cbvmptv 5261 | . . 3 ⊢ (𝑦 ∈ ℝ ↦ -𝑦) = (𝑥 ∈ ℝ ↦ -𝑥) |
30 | 12 | simpri 486 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦) |
31 | 29, 30, 1 | 3eqtr4i 2770 | . 2 ⊢ ◡𝐹 = 𝐹 |
32 | 27, 31 | pm3.2i 471 | 1 ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ∀wral 3061 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5675 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 ℂcc 11107 ℝcr 11108 < clt 11247 -cneg 11444 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 |
This theorem is referenced by: infrenegsup 12196 |
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