irrdiff - Mathbox for Jim Kingdon

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Theorem irrdiff 37292

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 37291	. . . 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 3208	. 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 11603	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
11		recn 11274	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
12		1cnd 11285	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℂ)
13	11, 12	negsubd 11653	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + -1) = (𝐴 − 1))
14		neg1lt0 12410	. . . . . . . . 9 ⊢ -1 < 0
15		0lt1 11812	. . . . . . . . 9 ⊢ 0 < 1
16		neg1rr 12408	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
17		0re 11292	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
18		1re 11290	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
19	16, 17, 18	lttri 11416	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
20	14, 15, 19	mp2an 691	. . . . . . . 8 ⊢ -1 < 1
21	16	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -1 ∈ ℝ)
22		1red 11291	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℝ)
23		id 22	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ)
24	21, 22, 23	ltadd2d 11446	. . . . . . . 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 5190	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < (𝐴 + 1))
27	10, 26	ltned 11426	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≠ (𝐴 + 1))
28	27	ad2antrr 725	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → (𝐴 − 1) ≠ (𝐴 + 1))
29		1z 12673	. . . . . . 7 ⊢ 1 ∈ ℤ
30		zq 13019	. . . . . . 7 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
31	29, 30	ax-mp 5	. . . . . 6 ⊢ 1 ∈ ℚ
32		qsubcl 13033	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 − 1) ∈ ℚ)
33	31, 32	mpan2 690	. . . . 5 ⊢ (𝐴 ∈ ℚ → (𝐴 − 1) ∈ ℚ)
34		qaddcl 13030	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 + 1) ∈ ℚ)
35	31, 34	mpan2 690	. . . . . 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 3008	. . . . . . 7 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (𝑞 ≠ 𝑟 ↔ (𝐴 − 1) ≠ (𝐴 + 1)))
40		oveq2 7456	. . . . . . . . . 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 6924	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − (𝐴 − 1))))
43		oveq2 7456	. . . . . . . . . 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 6924	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − (𝐴 + 1))))
46	42, 45	neeq12d 3008	. . . . . . 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 3645	. . . . 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 685	. . . 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 2955	. . . . 5 ⊢ ¬ (abs‘1) ≠ (abs‘1)
52	11, 12	nncand 11652	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − 1)) = 1)
53	52	fveq2d 6924	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 − 1))) = (abs‘1))
54	11, 12	subnegd 11654	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 − -1) = (𝐴 + 1))
55	54	oveq2d 7464	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − -1)) = (𝐴 − (𝐴 + 1)))
56	21	recnd 11318	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -1 ∈ ℂ)
57	11, 56	nncand 11652	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − -1)) = -1)
58	55, 57	eqtr3d 2782	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 + 1)) = -1)
59	58	fveq2d 6924	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 + 1))) = (abs‘-1))
60	12	absnegd 15498	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (abs‘-1) = (abs‘1))
61	59, 60	eqtrd 2780	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 + 1))) = (abs‘1))
62	53, 61	neeq12d 3008	. . . . 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 725	. . 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 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 < clt 11324 − cmin 11520 -cneg 11521 ℤcz 12639 ℚcq 13013 abscabs 15283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285

This theorem is referenced by: (None)

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