sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12094

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 12112 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12093, 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 8869	. . . . . 6
2		breq2 3986	. . . . . . . . 9
3	2	imbi1d 230	. . . . . . . 8
4	3	ralbidv 2466	. . . . . . 7
5		breq2 3986	. . . . . . . . 9
6	5	imbi1d 230	. . . . . . . 8
7	6	ralbidv 2466	. . . . . . 7
8		breq2 3986	. . . . . . . . 9
9	8	imbi1d 230	. . . . . . . 8
10	9	ralbidv 2466	. . . . . . 7
11		nnnlt1 8883	. . . . . . . . 9
12	11	pm2.21d 609	. . . . . . . 8
13	12	rgen 2519	. . . . . . 7
14		nnrp 9599	. . . . . . . . . . . . . 14
15		rphalflt 9619	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 3985	. . . . . . . . . . . . . . . 16
18		oveq2 5850	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2355	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2466	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 233	. . . . . . . . . . . . . . 15
22	21	rspcv 2826	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 109	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9196	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 481	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 8865	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 480	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 8930	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 8886	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 480	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 8950	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8715	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2201	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 520	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 519	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12093	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 113	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 111	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5849	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2355	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2826	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2394	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 114	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2393	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2546	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 5849	. . . . . . . . . . . . 13
54	53	neeq2d 2355	. . . . . . . . . . . 12
55	54	cbvralv 2692	. . . . . . . . . . 11
56	52, 55	syl6ibr 161	. . . . . . . . . 10
57		oveq2 5850	. . . . . . . . . . . . 13
58	57	neeq2d 2355	. . . . . . . . . . . 12
59	58	ralbidv 2466	. . . . . . . . . . 11
60	59	ceqsralv 2757	. . . . . . . . . 10
61	56, 60	sylibrd 168	. . . . . . . . 9
62	61	ancld 323	. . . . . . . 8 $/\$
63		nnleltp1 9250	. . . . . . . . . . . . . 14 $/\$
64		nnz 9210	. . . . . . . . . . . . . . 15
65		nnz 9210	. . . . . . . . . . . . . . 15
66		zleloe 9238	. . . . . . . . . . . . . . 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 700	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	bitrdi 195	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2462	. . . . . . . . 9 $/\$
74		r19.26 2592	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 195	. . . . . . . 8 $/\$
76	62, 75	sylibrd 168	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 8873	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 8864	. . . . . 6
80	79	ltp1d 8825	. . . . 5
81		breq1 3985	. . . . . . 7
82		df-ne 2337	. . . . . . . . . 10
83	58, 82	bitrdi 195	. . . . . . . . 9
84	83	ralbidv 2466	. . . . . . . 8
85		ralnex 2454	. . . . . . . 8
86	84, 85	bitrdi 195	. . . . . . 7
87	81, 86	imbi12d 233	. . . . . 6
88	87	rspcv 2826	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2558	. . 3
91		elq 9560	. . . 4
92		rexcom 2630	. . . 4
93	91, 92	bitri 183	. . 3
94	90, 93	mtbir 661	. 2
95	94	nelir 2434	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1343

wcel 2136

wne 2336

wnel 2431

wral 2444

wrex 2445 class class class wbr 3982

cfv 5188 (class class class)co 5842

cc 7751

cc0 7753

c1 7754

caddc 7756

clt 7933

cle 7934 # cap 8479

cdiv 8568

cn 8857

c2 8908

cz 9191

cq 9557

crp 9589

csqrt 10938

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-seqfrec 10381 df-exp 10455 df-rsqrt 10940

This theorem is referenced by: sqrt2irr0 12096

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