sqrt2irr - Metamath Proof Explorer

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Theorem sqrt2irr 14979

Description: The square root of 2 is irrational. See zsqrtelqelz 15466 for a generalization to all non-square integers. The proof's core is proven in sqrt2irrlem 14977, which shows that if

, then

and

are even, so

and

are smaller representatives, which is absurd. An older version of this proof was included in The Seventeen Provers of the World compiled by Freek Wiedijk. It is also the first of the "top 100" mathematical theorems whose formalization is tracked by Freek Wiedijk on his Formalizing 100 Theorems page at http://www.cs.ru.nl/~freek/100/. (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 11032	. . . . . 6
2		breq2 4657	. . . . . . . . 9
3	2	imbi1d 331	. . . . . . . 8
4	3	ralbidv 2986	. . . . . . 7
5		breq2 4657	. . . . . . . . 9
6	5	imbi1d 331	. . . . . . . 8
7	6	ralbidv 2986	. . . . . . 7
8		breq2 4657	. . . . . . . . 9
9	8	imbi1d 331	. . . . . . . 8
10	9	ralbidv 2986	. . . . . . 7
11		nnnlt1 11050	. . . . . . . . 9
12	11	pm2.21d 118	. . . . . . . 8
13	12	rgen 2922	. . . . . . 7
14		nnrp 11842	. . . . . . . . . . . . . 14
15		rphalflt 11860	. . . . . . . . . . . . . 14
16	14, 15	syl 17	. . . . . . . . . . . . 13
17		breq1 4656	. . . . . . . . . . . . . . . 16
18		oveq2 6658	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2854	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2986	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 334	. . . . . . . . . . . . . . 15
22	21	rspcv 3305	. . . . . . . . . . . . . 14
23	22	com13 88	. . . . . . . . . . . . 13
24	16, 23	syl 17	. . . . . . . . . . . 12
25		simpr 477	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 11382	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 11028	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 11093	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnne0 11053	. . . . . . . . . . . . . . . . . . 19
32	31	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
33		2ne0 11113	. . . . . . . . . . . . . . . . . . 19
34	33	a1i 11	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
35	27, 29, 30, 32, 34	divcan7d 10829	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2659	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 792	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 790	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 14977	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 479	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 475	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 6657	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2854	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 3305	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 17	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 59	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2810	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 450	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2809	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2969	. . . . . . . . . . . 12
52	24, 51	syld 47	. . . . . . . . . . 11
53		oveq1 6657	. . . . . . . . . . . . 13
54	53	neeq2d 2854	. . . . . . . . . . . 12
55	54	cbvralv 3171	. . . . . . . . . . 11
56	52, 55	syl6ibr 242	. . . . . . . . . 10
57		oveq2 6658	. . . . . . . . . . . . 13
58	57	neeq2d 2854	. . . . . . . . . . . 12
59	58	ralbidv 2986	. . . . . . . . . . 11
60	59	ceqsralv 3234	. . . . . . . . . 10
61	56, 60	sylibrd 249	. . . . . . . . 9
62	61	ancld 576	. . . . . . . 8 $/\$
63		nnleltp1 11432	. . . . . . . . . . . . . 14 $/\$
64		nnre 11027	. . . . . . . . . . . . . . 15
65		nnre 11027	. . . . . . . . . . . . . . 15
66		leloe 10124	. . . . . . . . . . . . . . 15 $/\$ $\/$
67	64, 65, 66	syl2an 494	. . . . . . . . . . . . . 14 $/\$ $\/$
68	63, 67	bitr3d 270	. . . . . . . . . . . . 13 $/\$ $\/$
69	68	ancoms 469	. . . . . . . . . . . 12 $/\$ $\/$
70	69	imbi1d 331	. . . . . . . . . . 11 $/\$ $\/$
71		jaob 822	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	syl6bb 276	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2985	. . . . . . . . 9 $/\$
74		r19.26 3064	. . . . . . . . 9 $/\$ $/\$
75	73, 74	syl6bb 276	. . . . . . . 8 $/\$
76	62, 75	sylibrd 249	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 11038	. . . . . 6
78	1, 77	syl 17	. . . . 5
79	65	ltp1d 10954	. . . . 5
80		breq1 4656	. . . . . . 7
81		df-ne 2795	. . . . . . . . . 10
82	58, 81	syl6bb 276	. . . . . . . . 9
83	82	ralbidv 2986	. . . . . . . 8
84		ralnex 2992	. . . . . . . 8
85	83, 84	syl6bb 276	. . . . . . 7
86	80, 85	imbi12d 334	. . . . . 6
87	86	rspcv 3305	. . . . 5
88	78, 79, 87	mp2d 49	. . . 4
89	88	nrex 3000	. . 3
90		elq 11790	. . . 4
91		rexcom 3099	. . . 4
92	90, 91	bitri 264	. . 3
93	89, 92	mtbir 313	. 2
94	93	nelir 2900	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

wnel 2897

wral 2912

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

clt 10074

cle 10075

cdiv 10684

cn 11020

c2 11070

cz 11377

cq 11788

crp 11832

csqrt 13973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976

This theorem is referenced by: nthruc 14981

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