sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 11829

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 11847 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 11828, 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 8725	. . . . . 6
2		breq2 3928	. . . . . . . . 9
3	2	imbi1d 230	. . . . . . . 8
4	3	ralbidv 2435	. . . . . . 7
5		breq2 3928	. . . . . . . . 9
6	5	imbi1d 230	. . . . . . . 8
7	6	ralbidv 2435	. . . . . . 7
8		breq2 3928	. . . . . . . . 9
9	8	imbi1d 230	. . . . . . . 8
10	9	ralbidv 2435	. . . . . . 7
11		nnnlt1 8739	. . . . . . . . 9
12	11	pm2.21d 608	. . . . . . . 8
13	12	rgen 2483	. . . . . . 7
14		nnrp 9444	. . . . . . . . . . . . . 14
15		rphalflt 9464	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 3927	. . . . . . . . . . . . . . . 16
18		oveq2 5775	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2325	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2435	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 233	. . . . . . . . . . . . . . 15
22	21	rspcv 2780	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 109	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9052	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 480	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 8721	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 479	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 8786	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 8742	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 479	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 8806	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8572	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2173	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 519	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 518	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 11828	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 113	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 111	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5774	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2325	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2780	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2364	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 114	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2363	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2510	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 5774	. . . . . . . . . . . . 13
54	53	neeq2d 2325	. . . . . . . . . . . 12
55	54	cbvralv 2652	. . . . . . . . . . 11
56	52, 55	syl6ibr 161	. . . . . . . . . 10
57		oveq2 5775	. . . . . . . . . . . . 13
58	57	neeq2d 2325	. . . . . . . . . . . 12
59	58	ralbidv 2435	. . . . . . . . . . 11
60	59	ceqsralv 2712	. . . . . . . . . 10
61	56, 60	sylibrd 168	. . . . . . . . 9
62	61	ancld 323	. . . . . . . 8 $/\$
63		nnleltp1 9106	. . . . . . . . . . . . . 14 $/\$
64		nnz 9066	. . . . . . . . . . . . . . 15
65		nnz 9066	. . . . . . . . . . . . . . 15
66		zleloe 9094	. . . . . . . . . . . . . . 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 2431	. . . . . . . . 9 $/\$
74		r19.26 2556	. . . . . . . . 9 $/\$ $/\$
75	73, 74	syl6bb 195	. . . . . . . 8 $/\$
76	62, 75	sylibrd 168	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 8729	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 8720	. . . . . 6
80	79	ltp1d 8681	. . . . 5
81		breq1 3927	. . . . . . 7
82		df-ne 2307	. . . . . . . . . 10
83	58, 82	syl6bb 195	. . . . . . . . 9
84	83	ralbidv 2435	. . . . . . . 8
85		ralnex 2424	. . . . . . . 8
86	84, 85	syl6bb 195	. . . . . . 7
87	81, 86	imbi12d 233	. . . . . 6
88	87	rspcv 2780	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2522	. . 3
91		elq 9407	. . . 4
92		rexcom 2593	. . . 4
93	91, 92	bitri 183	. . 3
94	90, 93	mtbir 660	. 2
95	94	nelir 2404	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 697

wceq 1331

wcel 1480

wne 2306

wnel 2401

wral 2414

wrex 2415 class class class wbr 3924

cfv 5118 (class class class)co 5767

cc 7611

cc0 7613

c1 7614

caddc 7616

clt 7793

cle 7794 # cap 8336

cdiv 8425

cn 8713

c2 8764

cz 9047

cq 9404

crp 9434

csqrt 10761

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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733

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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-seqfrec 10212 df-exp 10286 df-rsqrt 10763

This theorem is referenced by: sqrt2irr0 11831

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