sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12852

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 12870 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12851, 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 9245	. . . . . 6
2		breq2 4112	. . . . . . . . 9
3	2	imbi1d 231	. . . . . . . 8
4	3	ralbidv 2542	. . . . . . 7
5		breq2 4112	. . . . . . . . 9
6	5	imbi1d 231	. . . . . . . 8
7	6	ralbidv 2542	. . . . . . 7
8		breq2 4112	. . . . . . . . 9
9	8	imbi1d 231	. . . . . . . 8
10	9	ralbidv 2542	. . . . . . 7
11		nnnlt1 9259	. . . . . . . . 9
12	11	pm2.21d 624	. . . . . . . 8
13	12	rgen 2595	. . . . . . 7
14		nnrp 9992	. . . . . . . . . . . . . 14
15		rphalflt 10012	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 4111	. . . . . . . . . . . . . . . 16
18		oveq2 6057	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2431	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2542	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 234	. . . . . . . . . . . . . . 15
22	21	rspcv 2916	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 110	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9578	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 9241	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 9306	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 9262	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 9326	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 9089	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2268	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 529	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 527	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12851	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 6056	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2431	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2916	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2470	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 115	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2469	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2622	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 6056	. . . . . . . . . . . . 13
54	53	neeq2d 2431	. . . . . . . . . . . 12
55	54	cbvralv 2777	. . . . . . . . . . 11
56	52, 55	imbitrrdi 162	. . . . . . . . . 10
57		oveq2 6057	. . . . . . . . . . . . 13
58	57	neeq2d 2431	. . . . . . . . . . . 12
59	58	ralbidv 2542	. . . . . . . . . . 11
60	59	ceqsralv 2844	. . . . . . . . . 10
61	56, 60	sylibrd 169	. . . . . . . . 9
62	61	ancld 325	. . . . . . . 8 $/\$
63		nnleltp1 9633	. . . . . . . . . . . . . 14 $/\$
64		nnz 9592	. . . . . . . . . . . . . . 15
65		nnz 9592	. . . . . . . . . . . . . . 15
66		zleloe 9620	. . . . . . . . . . . . . . 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 718	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	bitrdi 196	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2538	. . . . . . . . 9 $/\$
74		r19.26 2669	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 196	. . . . . . . 8 $/\$
76	62, 75	sylibrd 169	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 9249	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 9240	. . . . . 6
80	79	ltp1d 9200	. . . . 5
81		breq1 4111	. . . . . . 7
82		df-ne 2413	. . . . . . . . . 10
83	58, 82	bitrdi 196	. . . . . . . . 9
84	83	ralbidv 2542	. . . . . . . 8
85		ralnex 2530	. . . . . . . 8
86	84, 85	bitrdi 196	. . . . . . 7
87	81, 86	imbi12d 234	. . . . . 6
88	87	rspcv 2916	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2634	. . 3
91		elq 9950	. . . 4
92		rexcom 2707	. . . 4
93	91, 92	bitri 184	. . 3
94	90, 93	mtbir 678	. 2
95	94	nelir 2510	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2203

wne 2412

wnel 2507

wral 2520

wrex 2521 class class class wbr 4108

cfv 5351 (class class class)co 6049

cc 8121

cc0 8123

c1 8124

caddc 8126

clt 8304

cle 8305 # cap 8851

cdiv 8942

cn 9233

c2 9284

cz 9573

cq 9947

crp 9982

csqrt 11674

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241 ax-arch 8242 ax-caucvg 8243

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-2 9292 df-3 9293 df-4 9294 df-n0 9493 df-z 9574 df-uz 9850 df-q 9948 df-rp 9983 df-seqfrec 10806 df-exp 10897 df-rsqrt 11676

This theorem is referenced by: sqrt2irr0 12854

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