sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12116

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 12134 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12115, 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 8890	. . . . . 6
2		breq2 3993	. . . . . . . . 9
3	2	imbi1d 230	. . . . . . . 8
4	3	ralbidv 2470	. . . . . . 7
5		breq2 3993	. . . . . . . . 9
6	5	imbi1d 230	. . . . . . . 8
7	6	ralbidv 2470	. . . . . . 7
8		breq2 3993	. . . . . . . . 9
9	8	imbi1d 230	. . . . . . . 8
10	9	ralbidv 2470	. . . . . . 7
11		nnnlt1 8904	. . . . . . . . 9
12	11	pm2.21d 614	. . . . . . . 8
13	12	rgen 2523	. . . . . . 7
14		nnrp 9620	. . . . . . . . . . . . . 14
15		rphalflt 9640	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 3992	. . . . . . . . . . . . . . . 16
18		oveq2 5861	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2359	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2470	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 233	. . . . . . . . . . . . . . 15
22	21	rspcv 2830	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 109	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9217	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 486	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 8886	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 485	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 8951	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 8907	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 485	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 8971	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8736	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2206	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 525	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 524	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12115	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 113	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 111	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5860	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2359	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2830	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2398	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 114	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2397	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2550	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 5860	. . . . . . . . . . . . 13
54	53	neeq2d 2359	. . . . . . . . . . . 12
55	54	cbvralv 2696	. . . . . . . . . . 11
56	52, 55	syl6ibr 161	. . . . . . . . . 10
57		oveq2 5861	. . . . . . . . . . . . 13
58	57	neeq2d 2359	. . . . . . . . . . . 12
59	58	ralbidv 2470	. . . . . . . . . . 11
60	59	ceqsralv 2761	. . . . . . . . . 10
61	56, 60	sylibrd 168	. . . . . . . . 9
62	61	ancld 323	. . . . . . . 8 $/\$
63		nnleltp1 9271	. . . . . . . . . . . . . 14 $/\$
64		nnz 9231	. . . . . . . . . . . . . . 15
65		nnz 9231	. . . . . . . . . . . . . . 15
66		zleloe 9259	. . . . . . . . . . . . . . 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 705	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	bitrdi 195	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2466	. . . . . . . . 9 $/\$
74		r19.26 2596	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 195	. . . . . . . 8 $/\$
76	62, 75	sylibrd 168	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 8894	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 8885	. . . . . 6
80	79	ltp1d 8846	. . . . 5
81		breq1 3992	. . . . . . 7
82		df-ne 2341	. . . . . . . . . 10
83	58, 82	bitrdi 195	. . . . . . . . 9
84	83	ralbidv 2470	. . . . . . . 8
85		ralnex 2458	. . . . . . . 8
86	84, 85	bitrdi 195	. . . . . . 7
87	81, 86	imbi12d 233	. . . . . 6
88	87	rspcv 2830	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2562	. . 3
91		elq 9581	. . . 4
92		rexcom 2634	. . . 4
93	91, 92	bitri 183	. . 3
94	90, 93	mtbir 666	. 2
95	94	nelir 2438	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141

wne 2340

wnel 2435

wral 2448

wrex 2449 class class class wbr 3989

cfv 5198 (class class class)co 5853

cc 7772

cc0 7774

c1 7775

caddc 7777

clt 7954

cle 7955 # cap 8500

cdiv 8589

cn 8878

c2 8929

cz 9212

cq 9578

crp 9610

csqrt 10960

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-seqfrec 10402 df-exp 10476 df-rsqrt 10962

This theorem is referenced by: sqrt2irr0 12118

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