sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12923

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 12941 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12922, 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 9299	. . . . . 6
2		breq2 4132	. . . . . . . . 9
3	2	imbi1d 231	. . . . . . . 8
4	3	ralbidv 2550	. . . . . . 7
5		breq2 4132	. . . . . . . . 9
6	5	imbi1d 231	. . . . . . . 8
7	6	ralbidv 2550	. . . . . . 7
8		breq2 4132	. . . . . . . . 9
9	8	imbi1d 231	. . . . . . . 8
10	9	ralbidv 2550	. . . . . . 7
11		nnnlt1 9313	. . . . . . . . 9
12	11	pm2.21d 628	. . . . . . . 8
13	12	rgen 2603	. . . . . . 7
14		nnrp 10047	. . . . . . . . . . . . . 14
15		rphalflt 10067	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 4131	. . . . . . . . . . . . . . . 16
18		oveq2 6087	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2439	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2550	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 234	. . . . . . . . . . . . . . 15
22	21	rspcv 2925	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 110	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9632	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 493	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 9295	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 492	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 9360	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 9316	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 492	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 9380	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 9143	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2274	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 533	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 531	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12922	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 6086	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2439	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2925	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2478	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 115	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2477	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2630	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 6086	. . . . . . . . . . . . 13
54	53	neeq2d 2439	. . . . . . . . . . . 12
55	54	cbvralv 2786	. . . . . . . . . . 11
56	52, 55	imbitrrdi 162	. . . . . . . . . 10
57		oveq2 6087	. . . . . . . . . . . . 13
58	57	neeq2d 2439	. . . . . . . . . . . 12
59	58	ralbidv 2550	. . . . . . . . . . 11
60	59	ceqsralv 2853	. . . . . . . . . 10
61	56, 60	sylibrd 169	. . . . . . . . 9
62	61	ancld 325	. . . . . . . 8 $/\$
63		nnleltp1 9687	. . . . . . . . . . . . . 14 $/\$
64		nnz 9646	. . . . . . . . . . . . . . 15
65		nnz 9646	. . . . . . . . . . . . . . 15
66		zleloe 9674	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	bitrdi 196	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2546	. . . . . . . . 9 $/\$
74		r19.26 2677	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 196	. . . . . . . 8 $/\$
76	62, 75	sylibrd 169	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 9303	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 9294	. . . . . 6
80	79	ltp1d 9254	. . . . 5
81		breq1 4131	. . . . . . 7
82		df-ne 2421	. . . . . . . . . 10
83	58, 82	bitrdi 196	. . . . . . . . 9
84	83	ralbidv 2550	. . . . . . . 8
85		ralnex 2538	. . . . . . . 8
86	84, 85	bitrdi 196	. . . . . . 7
87	81, 86	imbi12d 234	. . . . . 6
88	87	rspcv 2925	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2642	. . 3
91		elq 10005	. . . 4
92		rexcom 2715	. . . 4
93	91, 92	bitri 184	. . 3
94	90, 93	mtbir 682	. 2
95	94	nelir 2518	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209

wne 2420

wnel 2515

wral 2528

wrex 2529 class class class wbr 4128

cfv 5375 (class class class)co 6079

cc 8171

cc0 8173

c1 8174

caddc 8176

clt 8354

cle 8355 # cap 8903

cdiv 8996

cn 9287

c2 9338

cz 9627

cq 10002

crp 10037

csqrt 11745

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-mulrcl 8272 ax-addcom 8273 ax-mulcom 8274 ax-addass 8275 ax-mulass 8276 ax-distr 8277 ax-i2m1 8278 ax-0lt1 8279 ax-1rid 8280 ax-0id 8281 ax-rnegex 8282 ax-precex 8283 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltwlin 8286 ax-pre-lttrn 8287 ax-pre-apti 8288 ax-pre-ltadd 8289 ax-pre-mulgt0 8290 ax-pre-mulext 8291 ax-arch 8292 ax-caucvg 8293

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-recs 6570 df-frec 6656 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-le 8360 df-sub 8493 df-neg 8494 df-reap 8897 df-ap 8904 df-div 8997 df-inn 9288 df-2 9346 df-3 9347 df-4 9348 df-n0 9547 df-z 9628 df-uz 9905 df-q 10003 df-rp 10038 df-seqfrec 10868 df-exp 10959 df-rsqrt 11747

This theorem is referenced by: sqrt2irr0 12925

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