Theorem negiso 9063

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.

Step	Hyp	Ref	Expression
1		negiso.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥)
2		simpr 110	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
3	2	renegcld 8487	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -𝑥 ∈ ℝ)
4		simpr 110	. . . . . . 7 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
5	4	renegcld 8487	. . . . . 6 ⊢ ((⊤ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ)
6		recn 8093	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
7		recn 8093	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
8		negcon2 8360	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))
9	6, 7, 8	syl2an 289	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))
10	9	adantl 277	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 = -𝑦 ↔ 𝑦 = -𝑥))
11	1, 3, 5, 10	f1ocnv2d 6173	. . . . 5 ⊢ (⊤ → (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)))
12	11	mptru 1382	. . . 4 ⊢ (𝐹:ℝ–1-1-onto→ℝ ∧ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦))
13	12	simpli 111	. . 3 ⊢ 𝐹:ℝ–1-1-onto→ℝ
14		simpl 109	. . . . . . . 8 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑧 ∈ ℝ)
15	14	recnd 8136	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑧 ∈ ℂ)
16	15	negcld 8405	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → -𝑧 ∈ ℂ)
17	7	adantl 277	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
18	17	negcld 8405	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℂ)
19		brcnvg 4877	. . . . . 6 ⊢ ((-𝑧 ∈ ℂ ∧ -𝑦 ∈ ℂ) → (-𝑧◡ < -𝑦 ↔ -𝑦 < -𝑧))
20	16, 18, 19	syl2anc 411	. . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑧◡ < -𝑦 ↔ -𝑦 < -𝑧))
21	1	a1i 9	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝐹 = (𝑥 ∈ ℝ ↦ -𝑥))
22		negeq 8300	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → -𝑥 = -𝑧)
23	22	adantl 277	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 = 𝑧) → -𝑥 = -𝑧)
24	21, 23, 14, 16	fvmptd 5683	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑧) = -𝑧)
25		negeq 8300	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → -𝑥 = -𝑦)
26	25	adantl 277	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 = 𝑦) → -𝑥 = -𝑦)
27		simpr 110	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
28	21, 26, 27, 18	fvmptd 5683	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐹‘𝑦) = -𝑦)
29	24, 28	breq12d 4072	. . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐹‘𝑧)◡ < (𝐹‘𝑦) ↔ -𝑧◡ < -𝑦))
30		ltneg 8570	. . . . 5 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ -𝑦 < -𝑧))
31	20, 29, 30	3bitr4rd 221	. . . 4 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦)))
32	31	rgen2a 2562	. . 3 ⊢ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦))
33		df-isom 5299	. . 3 ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ↔ (𝐹:ℝ–1-1-onto→ℝ ∧ ∀𝑧 ∈ ℝ ∀𝑦 ∈ ℝ (𝑧 < 𝑦 ↔ (𝐹‘𝑧)◡ < (𝐹‘𝑦))))
34	13, 32, 33	mpbir2an 945	. 2 ⊢ 𝐹 Isom < , ◡ < (ℝ, ℝ)
35		negeq 8300	. . . 4 ⊢ (𝑦 = 𝑥 → -𝑦 = -𝑥)
36	35	cbvmptv 4156	. . 3 ⊢ (𝑦 ∈ ℝ ↦ -𝑦) = (𝑥 ∈ ℝ ↦ -𝑥)
37	12	simpri 113	. . 3 ⊢ ◡𝐹 = (𝑦 ∈ ℝ ↦ -𝑦)
38	36, 37, 1	3eqtr4i 2238	. 2 ⊢ ◡𝐹 = 𝐹
39	34, 38	pm3.2i 272	1 ⊢ (𝐹 Isom < , ◡ < (ℝ, ℝ) ∧ ◡𝐹 = 𝐹)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373  ⊤wtru 1374   ∈ wcel 2178  ∀wral 2486   class class class wbr 4059   ↦ cmpt 4121  ^◡ccnv 4692  –1-1-onto→wf1o 5289  ‘cfv 5290   Isom wiso 5291  ℂcc 7958  ℝcr 7959   < clt 8142  -cneg 8279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-sub 8280  df-neg 8281

This theorem is referenced by:  infrenegsupex  9750