irrdiff - Mathbox for Jim Kingdon

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Theorem irrdiff 37339

Description: The irrationals are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 19-May-2024.)

**Assertion**
Ref	Expression
irrdiff	⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℚ ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))))

Distinct variable group: 𝐴,𝑞,𝑟

**Proof of Theorem irrdiff**
Step	Hyp	Ref	Expression
1		simplll 774	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → 𝐴 ∈ ℝ)
2		simpllr 775	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → ¬ 𝐴 ∈ ℚ)
3		simplrl 776	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → 𝑞 ∈ ℚ)
4		simplrr 777	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → 𝑟 ∈ ℚ)
5		simpr 484	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → 𝑞 ≠ 𝑟)
6	1, 2, 3, 4, 5	irrdifflemf 37338	. . . 4 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))
7	6	ex 412	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) → (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))))
8	7	ralrimivva 3173	. 2 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) → ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))))
9		simplr 768	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))))
10		peano2rem 11420	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
11		recn 11088	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
12		1cnd 11099	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℂ)
13	11, 12	negsubd 11470	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + -1) = (𝐴 − 1))
14		neg1lt0 12105	. . . . . . . . 9 ⊢ -1 < 0
15		0lt1 11631	. . . . . . . . 9 ⊢ 0 < 1
16		neg1rr 12103	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
17		0re 11106	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
18		1re 11104	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
19	16, 17, 18	lttri 11231	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
20	14, 15, 19	mp2an 692	. . . . . . . 8 ⊢ -1 < 1
21	16	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -1 ∈ ℝ)
22		1red 11105	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℝ)
23		id 22	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ)
24	21, 22, 23	ltadd2d 11261	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (-1 < 1 ↔ (𝐴 + -1) < (𝐴 + 1)))
25	20, 24	mpbii 233	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + -1) < (𝐴 + 1))
26	13, 25	eqbrtrrd 5113	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < (𝐴 + 1))
27	10, 26	ltned 11241	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≠ (𝐴 + 1))
28	27	ad2antrr 726	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → (𝐴 − 1) ≠ (𝐴 + 1))
29		1z 12494	. . . . . . 7 ⊢ 1 ∈ ℤ
30		zq 12844	. . . . . . 7 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
31	29, 30	ax-mp 5	. . . . . 6 ⊢ 1 ∈ ℚ
32		qsubcl 12858	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 − 1) ∈ ℚ)
33	31, 32	mpan2 691	. . . . 5 ⊢ (𝐴 ∈ ℚ → (𝐴 − 1) ∈ ℚ)
34		qaddcl 12855	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 + 1) ∈ ℚ)
35	31, 34	mpan2 691	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (𝐴 + 1) ∈ ℚ)
36	35	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → (𝐴 + 1) ∈ ℚ)
37		simpl 482	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → 𝑞 = (𝐴 − 1))
38		simpr 484	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → 𝑟 = (𝐴 + 1))
39	37, 38	neeq12d 2987	. . . . . . 7 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (𝑞 ≠ 𝑟 ↔ (𝐴 − 1) ≠ (𝐴 + 1)))
40		oveq2 7349	. . . . . . . . . 10 ⊢ (𝑞 = (𝐴 − 1) → (𝐴 − 𝑞) = (𝐴 − (𝐴 − 1)))
41	40	adantr 480	. . . . . . . . 9 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (𝐴 − 𝑞) = (𝐴 − (𝐴 − 1)))
42	41	fveq2d 6821	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − (𝐴 − 1))))
43		oveq2 7349	. . . . . . . . . 10 ⊢ (𝑟 = (𝐴 + 1) → (𝐴 − 𝑟) = (𝐴 − (𝐴 + 1)))
44	43	adantl 481	. . . . . . . . 9 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (𝐴 − 𝑟) = (𝐴 − (𝐴 + 1)))
45	44	fveq2d 6821	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − (𝐴 + 1))))
46	42, 45	neeq12d 2987	. . . . . . 7 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → ((abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)) ↔ (abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1)))))
47	39, 46	imbi12d 344	. . . . . 6 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → ((𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))) ↔ ((𝐴 − 1) ≠ (𝐴 + 1) → (abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1))))))
48	47	rspc2gv 3585	. . . . 5 ⊢ (((𝐴 − 1) ∈ ℚ ∧ (𝐴 + 1) ∈ ℚ) → (∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))) → ((𝐴 − 1) ≠ (𝐴 + 1) → (abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1))))))
49	33, 36, 48	syl2an2 686	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → (∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))) → ((𝐴 − 1) ≠ (𝐴 + 1) → (abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1))))))
50	9, 28, 49	mp2d 49	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → (abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1))))
51		neirr 2935	. . . . 5 ⊢ ¬ (abs‘1) ≠ (abs‘1)
52	11, 12	nncand 11469	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − 1)) = 1)
53	52	fveq2d 6821	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 − 1))) = (abs‘1))
54	11, 12	subnegd 11471	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 − -1) = (𝐴 + 1))
55	54	oveq2d 7357	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − -1)) = (𝐴 − (𝐴 + 1)))
56	21	recnd 11132	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -1 ∈ ℂ)
57	11, 56	nncand 11469	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − -1)) = -1)
58	55, 57	eqtr3d 2767	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 + 1)) = -1)
59	58	fveq2d 6821	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 + 1))) = (abs‘-1))
60	12	absnegd 15351	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (abs‘-1) = (abs‘1))
61	59, 60	eqtrd 2765	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 + 1))) = (abs‘1))
62	53, 61	neeq12d 2987	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1))) ↔ (abs‘1) ≠ (abs‘1)))
63	51, 62	mtbiri 327	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ (abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1))))
64	63	ad2antrr 726	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → ¬ (abs‘(𝐴 − (𝐴 − 1))) ≠ (abs‘(𝐴 − (𝐴 + 1))))
65	50, 64	pm2.65da 816	. 2 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) → ¬ 𝐴 ∈ ℚ)
66	8, 65	impbida 800	1 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℚ ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 ∀wral 3045 class class class wbr 5089 ‘cfv 6477 (class class class)co 7341 ℝcr 10997 0cc0 10998 1c1 10999 + caddc 11001 < clt 11138 − cmin 11336 -cneg 11337 ℤcz 12460 ℚcq 12838 abscabs 15133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-sup 9321 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-n0 12374 df-z 12461 df-uz 12725 df-q 12839 df-rp 12883 df-seq 13901 df-exp 13961 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135

This theorem is referenced by: (None)

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