sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12303

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 12321 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12302, 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 8996	. . . . . 6
2		breq2 4034	. . . . . . . . 9
3	2	imbi1d 231	. . . . . . . 8
4	3	ralbidv 2494	. . . . . . 7
5		breq2 4034	. . . . . . . . 9
6	5	imbi1d 231	. . . . . . . 8
7	6	ralbidv 2494	. . . . . . 7
8		breq2 4034	. . . . . . . . 9
9	8	imbi1d 231	. . . . . . . 8
10	9	ralbidv 2494	. . . . . . 7
11		nnnlt1 9010	. . . . . . . . 9
12	11	pm2.21d 620	. . . . . . . 8
13	12	rgen 2547	. . . . . . 7
14		nnrp 9732	. . . . . . . . . . . . . 14
15		rphalflt 9752	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 4033	. . . . . . . . . . . . . . . 16
18		oveq2 5927	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2383	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2494	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 234	. . . . . . . . . . . . . . 15
22	21	rspcv 2861	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 110	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9325	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 8992	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 9057	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 9013	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 9077	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8840	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2229	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 528	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 527	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12302	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5926	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2383	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2861	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2422	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 115	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2421	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2574	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 5926	. . . . . . . . . . . . 13
54	53	neeq2d 2383	. . . . . . . . . . . 12
55	54	cbvralv 2726	. . . . . . . . . . 11
56	52, 55	imbitrrdi 162	. . . . . . . . . 10
57		oveq2 5927	. . . . . . . . . . . . 13
58	57	neeq2d 2383	. . . . . . . . . . . 12
59	58	ralbidv 2494	. . . . . . . . . . 11
60	59	ceqsralv 2791	. . . . . . . . . 10
61	56, 60	sylibrd 169	. . . . . . . . 9
62	61	ancld 325	. . . . . . . 8 $/\$
63		nnleltp1 9379	. . . . . . . . . . . . . 14 $/\$
64		nnz 9339	. . . . . . . . . . . . . . 15
65		nnz 9339	. . . . . . . . . . . . . . 15
66		zleloe 9367	. . . . . . . . . . . . . . 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 711	. . . . . . . . . . 11 $\/$ $/\$
72	70, 71	bitrdi 196	. . . . . . . . . 10 $/\$ $/\$
73	72	ralbidva 2490	. . . . . . . . 9 $/\$
74		r19.26 2620	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 196	. . . . . . . 8 $/\$
76	62, 75	sylibrd 169	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 9000	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 8991	. . . . . 6
80	79	ltp1d 8951	. . . . 5
81		breq1 4033	. . . . . . 7
82		df-ne 2365	. . . . . . . . . 10
83	58, 82	bitrdi 196	. . . . . . . . 9
84	83	ralbidv 2494	. . . . . . . 8
85		ralnex 2482	. . . . . . . 8
86	84, 85	bitrdi 196	. . . . . . 7
87	81, 86	imbi12d 234	. . . . . 6
88	87	rspcv 2861	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2586	. . 3
91		elq 9690	. . . 4
92		rexcom 2658	. . . 4
93	91, 92	bitri 184	. . 3
94	90, 93	mtbir 672	. 2
95	94	nelir 2462	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2164

wne 2364

wnel 2459

wral 2472

wrex 2473 class class class wbr 4030

cfv 5255 (class class class)co 5919

cc 7872

cc0 7874

c1 7875

caddc 7877

clt 8056

cle 8057 # cap 8602

cdiv 8693

cn 8984

c2 9035

cz 9320

cq 9687

crp 9722

csqrt 11143

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-seqfrec 10522 df-exp 10613 df-rsqrt 11145

This theorem is referenced by: sqrt2irr0 12305

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