sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12757

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 12775 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12756, 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 9160	. . . . . 6
2		breq2 4093	. . . . . . . . 9
3	2	imbi1d 231	. . . . . . . 8
4	3	ralbidv 2531	. . . . . . 7
5		breq2 4093	. . . . . . . . 9
6	5	imbi1d 231	. . . . . . . 8
7	6	ralbidv 2531	. . . . . . 7
8		breq2 4093	. . . . . . . . 9
9	8	imbi1d 231	. . . . . . . 8
10	9	ralbidv 2531	. . . . . . 7
11		nnnlt1 9174	. . . . . . . . 9
12	11	pm2.21d 624	. . . . . . . 8
13	12	rgen 2584	. . . . . . 7
14		nnrp 9903	. . . . . . . . . . . . . 14
15		rphalflt 9923	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 4092	. . . . . . . . . . . . . . . 16
18		oveq2 6031	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2420	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2531	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 234	. . . . . . . . . . . . . . 15
22	21	rspcv 2905	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 110	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9489	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 9156	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 9221	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 9177	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 9241	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 9004	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2266	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 529	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 527	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12756	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 6030	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2420	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2905	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2459	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 115	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2458	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2611	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 6030	. . . . . . . . . . . . 13
54	53	neeq2d 2420	. . . . . . . . . . . 12
55	54	cbvralv 2766	. . . . . . . . . . 11
56	52, 55	imbitrrdi 162	. . . . . . . . . 10
57		oveq2 6031	. . . . . . . . . . . . 13
58	57	neeq2d 2420	. . . . . . . . . . . 12
59	58	ralbidv 2531	. . . . . . . . . . 11
60	59	ceqsralv 2833	. . . . . . . . . 10
61	56, 60	sylibrd 169	. . . . . . . . 9
62	61	ancld 325	. . . . . . . 8 $/\$
63		nnleltp1 9544	. . . . . . . . . . . . . 14 $/\$
64		nnz 9503	. . . . . . . . . . . . . . 15
65		nnz 9503	. . . . . . . . . . . . . . 15
66		zleloe 9531	. . . . . . . . . . . . . . 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 717	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	bitrdi 196	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2527	. . . . . . . . 9 $/\$
74		r19.26 2658	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 196	. . . . . . . 8 $/\$
76	62, 75	sylibrd 169	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 9164	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 9155	. . . . . 6
80	79	ltp1d 9115	. . . . 5
81		breq1 4092	. . . . . . 7
82		df-ne 2402	. . . . . . . . . 10
83	58, 82	bitrdi 196	. . . . . . . . 9
84	83	ralbidv 2531	. . . . . . . 8
85		ralnex 2519	. . . . . . . 8
86	84, 85	bitrdi 196	. . . . . . 7
87	81, 86	imbi12d 234	. . . . . 6
88	87	rspcv 2905	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2623	. . 3
91		elq 9861	. . . 4
92		rexcom 2696	. . . 4
93	91, 92	bitri 184	. . 3
94	90, 93	mtbir 677	. 2
95	94	nelir 2499	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 715

wceq 1397

wcel 2201

wne 2401

wnel 2496

wral 2509

wrex 2510 class class class wbr 4089

cfv 5328 (class class class)co 6023

cc 8035

cc0 8037

c1 8038

caddc 8040

clt 8219

cle 8220 # cap 8766

cdiv 8857

cn 9148

c2 9199

cz 9484

cq 9858

crp 9893

csqrt 11579

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-mulrcl 8136 ax-addcom 8137 ax-mulcom 8138 ax-addass 8139 ax-mulass 8140 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-1rid 8144 ax-0id 8145 ax-rnegex 8146 ax-precex 8147 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153 ax-pre-mulgt0 8154 ax-pre-mulext 8155 ax-arch 8156 ax-caucvg 8157

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-po 4395 df-iso 4396 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-recs 6476 df-frec 6562 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-reap 8760 df-ap 8767 df-div 8858 df-inn 9149 df-2 9207 df-3 9208 df-4 9209 df-n0 9408 df-z 9485 df-uz 9761 df-q 9859 df-rp 9894 df-seqfrec 10716 df-exp 10807 df-rsqrt 11581

This theorem is referenced by: sqrt2irr0 12759

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