sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 11840

Description: The square root of 2 is not rational. That is, for any rational number,

does not equal it. However, if we were to say "the square root of 2 is irrational" that would mean something stronger: "for any rational number,

is apart from it" (the two statements are equivalent given excluded middle). See sqrt2irrap 11858 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 11839, which shows that if , then and are even, so and are smaller representatives, which is absurd. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

**Assertion**
Ref	Expression
sqrt2irr

Proof of Theorem sqrt2irr Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2nn 8732	. . . . . 6
2		breq2 3933	. . . . . . . . 9
3	2	imbi1d 230	. . . . . . . 8
4	3	ralbidv 2437	. . . . . . 7
5		breq2 3933	. . . . . . . . 9
6	5	imbi1d 230	. . . . . . . 8
7	6	ralbidv 2437	. . . . . . 7
8		breq2 3933	. . . . . . . . 9
9	8	imbi1d 230	. . . . . . . 8
10	9	ralbidv 2437	. . . . . . 7
11		nnnlt1 8746	. . . . . . . . 9
12	11	pm2.21d 608	. . . . . . . 8
13	12	rgen 2485	. . . . . . 7
14		nnrp 9451	. . . . . . . . . . . . . 14
15		rphalflt 9471	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 3932	. . . . . . . . . . . . . . . 16
18		oveq2 5782	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2327	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2437	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 233	. . . . . . . . . . . . . . 15
22	21	rspcv 2785	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 109	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9059	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 480	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 8728	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 479	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 8793	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 8749	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 479	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 8813	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8579	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2175	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 519	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 518	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 11839	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 113	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 111	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5781	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2327	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2785	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2366	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 114	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2365	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2512	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 5781	. . . . . . . . . . . . 13
54	53	neeq2d 2327	. . . . . . . . . . . 12
55	54	cbvralv 2654	. . . . . . . . . . 11
56	52, 55	syl6ibr 161	. . . . . . . . . 10
57		oveq2 5782	. . . . . . . . . . . . 13
58	57	neeq2d 2327	. . . . . . . . . . . 12
59	58	ralbidv 2437	. . . . . . . . . . 11
60	59	ceqsralv 2717	. . . . . . . . . 10
61	56, 60	sylibrd 168	. . . . . . . . 9
62	61	ancld 323	. . . . . . . 8 $/\$
63		nnleltp1 9113	. . . . . . . . . . . . . 14 $/\$
64		nnz 9073	. . . . . . . . . . . . . . 15
65		nnz 9073	. . . . . . . . . . . . . . 15
66		zleloe 9101	. . . . . . . . . . . . . . 15 $/\$ $\/$
67	64, 65, 66	syl2an 287	. . . . . . . . . . . . . 14 $/\$ $\/$
68	63, 67	bitr3d 189	. . . . . . . . . . . . 13 $/\$ $\/$
69	68	ancoms 266	. . . . . . . . . . . 12 $/\$ $\/$
70	69	imbi1d 230	. . . . . . . . . . 11 $/\$ $\/$
71		jaob 699	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	syl6bb 195	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2433	. . . . . . . . 9 $/\$
74		r19.26 2558	. . . . . . . . 9 $/\$ $/\$
75	73, 74	syl6bb 195	. . . . . . . 8 $/\$
76	62, 75	sylibrd 168	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 8736	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 8727	. . . . . 6
80	79	ltp1d 8688	. . . . 5
81		breq1 3932	. . . . . . 7
82		df-ne 2309	. . . . . . . . . 10
83	58, 82	syl6bb 195	. . . . . . . . 9
84	83	ralbidv 2437	. . . . . . . 8
85		ralnex 2426	. . . . . . . 8
86	84, 85	syl6bb 195	. . . . . . 7
87	81, 86	imbi12d 233	. . . . . 6
88	87	rspcv 2785	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2524	. . 3
91		elq 9414	. . . 4
92		rexcom 2595	. . . 4
93	91, 92	bitri 183	. . 3
94	90, 93	mtbir 660	. 2
95	94	nelir 2406	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697

wceq 1331

wcel 1480

wne 2308

wnel 2403

wral 2416

wrex 2417 class class class wbr 3929

cfv 5123 (class class class)co 5774

cc 7618

cc0 7620

c1 7621

caddc 7623

clt 7800

cle 7801 # cap 8343

cdiv 8432

cn 8720

c2 8771

cz 9054

cq 9411

crp 9441

csqrt 10768

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-seqfrec 10219 df-exp 10293 df-rsqrt 10770

This theorem is referenced by: sqrt2irr0 11842

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