sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12161

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 12179 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12160, 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 8930	. . . . . 6
2		breq2 4007	. . . . . . . . 9
3	2	imbi1d 231	. . . . . . . 8
4	3	ralbidv 2477	. . . . . . 7
5		breq2 4007	. . . . . . . . 9
6	5	imbi1d 231	. . . . . . . 8
7	6	ralbidv 2477	. . . . . . 7
8		breq2 4007	. . . . . . . . 9
9	8	imbi1d 231	. . . . . . . 8
10	9	ralbidv 2477	. . . . . . 7
11		nnnlt1 8944	. . . . . . . . 9
12	11	pm2.21d 619	. . . . . . . 8
13	12	rgen 2530	. . . . . . 7
14		nnrp 9662	. . . . . . . . . . . . . 14
15		rphalflt 9682	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 4006	. . . . . . . . . . . . . . . 16
18		oveq2 5882	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2366	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2477	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 234	. . . . . . . . . . . . . . 15
22	21	rspcv 2837	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 110	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9257	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 8926	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 8991	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 8947	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 9011	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8775	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2213	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 528	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 527	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12160	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5881	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2366	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2837	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2405	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 115	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2404	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2557	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 5881	. . . . . . . . . . . . 13
54	53	neeq2d 2366	. . . . . . . . . . . 12
55	54	cbvralv 2703	. . . . . . . . . . 11
56	52, 55	imbitrrdi 162	. . . . . . . . . 10
57		oveq2 5882	. . . . . . . . . . . . 13
58	57	neeq2d 2366	. . . . . . . . . . . 12
59	58	ralbidv 2477	. . . . . . . . . . 11
60	59	ceqsralv 2768	. . . . . . . . . 10
61	56, 60	sylibrd 169	. . . . . . . . 9
62	61	ancld 325	. . . . . . . 8 $/\$
63		nnleltp1 9311	. . . . . . . . . . . . . 14 $/\$
64		nnz 9271	. . . . . . . . . . . . . . 15
65		nnz 9271	. . . . . . . . . . . . . . 15
66		zleloe 9299	. . . . . . . . . . . . . . 15 $/\$ $\/$
67	64, 65, 66	syl2an 289	. . . . . . . . . . . . . 14 $/\$ $\/$
68	63, 67	bitr3d 190	. . . . . . . . . . . . 13 $/\$ $\/$
69	68	ancoms 268	. . . . . . . . . . . 12 $/\$ $\/$
70	69	imbi1d 231	. . . . . . . . . . 11 $/\$ $\/$
71		jaob 710	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	bitrdi 196	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2473	. . . . . . . . 9 $/\$
74		r19.26 2603	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 196	. . . . . . . 8 $/\$
76	62, 75	sylibrd 169	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 8934	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 8925	. . . . . 6
80	79	ltp1d 8886	. . . . 5
81		breq1 4006	. . . . . . 7
82		df-ne 2348	. . . . . . . . . 10
83	58, 82	bitrdi 196	. . . . . . . . 9
84	83	ralbidv 2477	. . . . . . . 8
85		ralnex 2465	. . . . . . . 8
86	84, 85	bitrdi 196	. . . . . . 7
87	81, 86	imbi12d 234	. . . . . 6
88	87	rspcv 2837	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2569	. . 3
91		elq 9621	. . . 4
92		rexcom 2641	. . . 4
93	91, 92	bitri 184	. . 3
94	90, 93	mtbir 671	. 2
95	94	nelir 2445	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708

wceq 1353

wcel 2148

wne 2347

wnel 2442

wral 2455

wrex 2456 class class class wbr 4003

cfv 5216 (class class class)co 5874

cc 7808

cc0 7810

c1 7811

caddc 7813

clt 7991

cle 7992 # cap 8537

cdiv 8628

cn 8918

c2 8969

cz 9252

cq 9618

crp 9652

csqrt 11004

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-seqfrec 10445 df-exp 10519 df-rsqrt 11006

This theorem is referenced by: sqrt2irr0 12163

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