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Theorem negiso 8805
 Description: Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
negiso.1 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)
Assertion
Ref Expression
negiso (𝐹 Isom < , < (ℝ, ℝ) ∧ 𝐹 = 𝐹)

Proof of Theorem negiso
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negiso.1 . . . . . 6 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)
2 simpr 109 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
32renegcld 8234 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ) → -𝑥 ∈ ℝ)
4 simpr 109 . . . . . . 7 ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
54renegcld 8234 . . . . . 6 ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ)
6 recn 7844 . . . . . . . 8 (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
7 recn 7844 . . . . . . . 8 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8 negcon2 8107 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦𝑦 = -𝑥))
96, 7, 8syl2an 287 . . . . . . 7 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = -𝑦𝑦 = -𝑥))
109adantl 275 . . . . . 6 ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 = -𝑦𝑦 = -𝑥))
111, 3, 5, 10f1ocnv2d 6014 . . . . 5 (⊤ → (𝐹:ℝ–1-1-onto→ℝ ∧ 𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)))
1211mptru 1341 . . . 4 (𝐹:ℝ–1-1-onto→ℝ ∧ 𝐹 = (𝑦 ∈ ℝ ↦ -𝑦))
1312simpli 110 . . 3 𝐹:ℝ–1-1-onto→ℝ
14 simpl 108 . . . . . . . 8 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑧 ∈ ℝ)
1514recnd 7885 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑧 ∈ ℂ)
1615negcld 8152 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → -𝑧 ∈ ℂ)
177adantl 275 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
1817negcld 8152 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℂ)
19 brcnvg 4760 . . . . . 6 ((-𝑧 ∈ ℂ ∧ -𝑦 ∈ ℂ) → (-𝑧 < -𝑦 ↔ -𝑦 < -𝑧))
2016, 18, 19syl2anc 409 . . . . 5 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑧 < -𝑦 ↔ -𝑦 < -𝑧))
211a1i 9 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥))
22 negeq 8047 . . . . . . . 8 (𝑥 = 𝑧 → -𝑥 = -𝑧)
2322adantl 275 . . . . . . 7 (((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 = 𝑧) → -𝑥 = -𝑧)
2421, 23, 14, 16fvmptd 5542 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐹𝑧) = -𝑧)
25 negeq 8047 . . . . . . . 8 (𝑥 = 𝑦 → -𝑥 = -𝑦)
2625adantl 275 . . . . . . 7 (((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 = 𝑦) → -𝑥 = -𝑦)
27 simpr 109 . . . . . . 7 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
2821, 26, 27, 18fvmptd 5542 . . . . . 6 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = -𝑦)
2924, 28breq12d 3974 . . . . 5 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹𝑧) < (𝐹𝑦) ↔ -𝑧 < -𝑦))
30 ltneg 8316 . . . . 5 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑦 < -𝑧))
3120, 29, 303bitr4rd 220 . . . 4 ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ (𝐹𝑧) < (𝐹𝑦)))
3231rgen2a 2508 . . 3 𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹𝑧) < (𝐹𝑦))
33 df-isom 5172 . . 3 (𝐹 Isom < , < (ℝ, ℝ) ↔ (𝐹:ℝ–1-1-onto→ℝ ∧ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹𝑧) < (𝐹𝑦))))
3413, 32, 33mpbir2an 927 . 2 𝐹 Isom < , < (ℝ, ℝ)
35 negeq 8047 . . . 4 (𝑦 = 𝑥 → -𝑦 = -𝑥)
3635cbvmptv 4056 . . 3 (𝑦 ∈ ℝ ↦ -𝑦) = (𝑥 ∈ ℝ ↦ -𝑥)
3712simpri 112 . . 3 𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)
3836, 37, 13eqtr4i 2185 . 2 𝐹 = 𝐹
3934, 38pm3.2i 270 1 (𝐹 Isom < , < (ℝ, ℝ) ∧ 𝐹 = 𝐹)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1332  ⊤wtru 1333   ∈ wcel 2125  ∀wral 2432   class class class wbr 3961   ↦ cmpt 4021  ◡ccnv 4578  –1-1-onto→wf1o 5162  ‘cfv 5163   Isom wiso 5164  ℂcc 7709  ℝcr 7710   < clt 7891  -cneg 8026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-addcom 7811  ax-addass 7813  ax-distr 7815  ax-i2m1 7816  ax-0id 7819  ax-rnegex 7820  ax-cnre 7822  ax-pre-ltadd 7827 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-pnf 7893  df-mnf 7894  df-ltxr 7896  df-sub 8027  df-neg 8028 This theorem is referenced by:  infrenegsupex  9484
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