irrdiff - Mathbox for Jim Kingdon

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Theorem irrdiff 37653

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 775	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → 𝐴 ∈ ℝ)
2		simpllr 776	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → ¬ 𝐴 ∈ ℚ)
3		simplrl 777	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ 𝑞 ≠ 𝑟) → 𝑞 ∈ ℚ)
4		simplrr 778	. . . . 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 37652	. . . 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 3181	. 2 ⊢ ((𝐴 ∈ ℝ ∧ ¬ 𝐴 ∈ ℚ) → ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟))))
9		simplr 769	. . . 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 11450	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
11		recn 11117	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
12		1cnd 11128	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℂ)
13	11, 12	negsubd 11500	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + -1) = (𝐴 − 1))
14		neg1lt0 12136	. . . . . . . . 9 ⊢ -1 < 0
15		0lt1 11661	. . . . . . . . 9 ⊢ 0 < 1
16		neg1rr 12134	. . . . . . . . . 10 ⊢ -1 ∈ ℝ
17		0re 11135	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
18		1re 11133	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
19	16, 17, 18	lttri 11261	. . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < 1) → -1 < 1)
20	14, 15, 19	mp2an 693	. . . . . . . 8 ⊢ -1 < 1
21	16	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → -1 ∈ ℝ)
22		1red 11134	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 1 ∈ ℝ)
23		id 22	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ)
24	21, 22, 23	ltadd2d 11291	. . . . . . . 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 5110	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < (𝐴 + 1))
27	10, 26	ltned 11271	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≠ (𝐴 + 1))
28	27	ad2antrr 727	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) ∧ 𝐴 ∈ ℚ) → (𝐴 − 1) ≠ (𝐴 + 1))
29		1z 12546	. . . . . . 7 ⊢ 1 ∈ ℤ
30		zq 12893	. . . . . . 7 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
31	29, 30	ax-mp 5	. . . . . 6 ⊢ 1 ∈ ℚ
32		qsubcl 12907	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 − 1) ∈ ℚ)
33	31, 32	mpan2 692	. . . . 5 ⊢ (𝐴 ∈ ℚ → (𝐴 − 1) ∈ ℚ)
34		qaddcl 12904	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 + 1) ∈ ℚ)
35	31, 34	mpan2 692	. . . . . 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 2994	. . . . . . 7 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (𝑞 ≠ 𝑟 ↔ (𝐴 − 1) ≠ (𝐴 + 1)))
40		oveq2 7366	. . . . . . . . . 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 6836	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − (𝐴 − 1))))
43		oveq2 7366	. . . . . . . . . 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 6836	. . . . . . . 8 ⊢ ((𝑞 = (𝐴 − 1) ∧ 𝑟 = (𝐴 + 1)) → (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − (𝐴 + 1))))
46	42, 45	neeq12d 2994	. . . . . . 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 3575	. . . . 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 687	. . . 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 2942	. . . . 5 ⊢ ¬ (abs‘1) ≠ (abs‘1)
52	11, 12	nncand 11499	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − 1)) = 1)
53	52	fveq2d 6836	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 − 1))) = (abs‘1))
54	11, 12	subnegd 11501	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → (𝐴 − -1) = (𝐴 + 1))
55	54	oveq2d 7374	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − -1)) = (𝐴 − (𝐴 + 1)))
56	21	recnd 11162	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -1 ∈ ℂ)
57	11, 56	nncand 11499	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 − -1)) = -1)
58	55, 57	eqtr3d 2774	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 − (𝐴 + 1)) = -1)
59	58	fveq2d 6836	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 + 1))) = (abs‘-1))
60	12	absnegd 15403	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (abs‘-1) = (abs‘1))
61	59, 60	eqtrd 2772	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (abs‘(𝐴 − (𝐴 + 1))) = (abs‘1))
62	53, 61	neeq12d 2994	. . . . 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 727	. . 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 817	. 2 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))) → ¬ 𝐴 ∈ ℚ)
66	8, 65	impbida 801	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 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 class class class wbr 5086 ‘cfv 6490 (class class class)co 7358 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 < clt 11168 − cmin 11366 -cneg 11367 ℤcz 12513 ℚcq 12887 abscabs 15185

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9346 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-z 12514 df-uz 12778 df-q 12888 df-rp 12932 df-seq 13953 df-exp 14013 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187

This theorem is referenced by: (None)

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