sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12330

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 12348 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12329, 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 9002	. . . . . 6
2		breq2 4037	. . . . . . . . 9
3	2	imbi1d 231	. . . . . . . 8
4	3	ralbidv 2497	. . . . . . 7
5		breq2 4037	. . . . . . . . 9
6	5	imbi1d 231	. . . . . . . 8
7	6	ralbidv 2497	. . . . . . 7
8		breq2 4037	. . . . . . . . 9
9	8	imbi1d 231	. . . . . . . 8
10	9	ralbidv 2497	. . . . . . 7
11		nnnlt1 9016	. . . . . . . . 9
12	11	pm2.21d 620	. . . . . . . 8
13	12	rgen 2550	. . . . . . 7
14		nnrp 9738	. . . . . . . . . . . . . 14
15		rphalflt 9758	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 4036	. . . . . . . . . . . . . . . 16
18		oveq2 5930	. . . . . . . . . . . . . . . . . 18
19	18	neeq2d 2386	. . . . . . . . . . . . . . . . 17
20	19	ralbidv 2497	. . . . . . . . . . . . . . . 16
21	17, 20	imbi12d 234	. . . . . . . . . . . . . . 15
22	21	rspcv 2864	. . . . . . . . . . . . . 14
23	22	com13 80	. . . . . . . . . . . . 13
24	16, 23	syl 14	. . . . . . . . . . . 12
25		simpr 110	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
26		zcn 9331	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 8998	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 9063	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 9019	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 9083	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8846	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2232	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 528	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 527	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12329	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5929	. . . . . . . . . . . . . . . . . . . . 21
43	42	neeq2d 2386	. . . . . . . . . . . . . . . . . . . 20
44	43	rspcv 2864	. . . . . . . . . . . . . . . . . . 19
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
47	46	necon2bd 2425	. . . . . . . . . . . . . . . 16 $/\$ $/\$
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 $/\$ $/\$
49	48	ex 115	. . . . . . . . . . . . . 14 $/\$
50	49	necon2ad 2424	. . . . . . . . . . . . 13 $/\$
51	50	ralrimdva 2577	. . . . . . . . . . . 12
52	24, 51	syld 45	. . . . . . . . . . 11
53		oveq1 5929	. . . . . . . . . . . . 13
54	53	neeq2d 2386	. . . . . . . . . . . 12
55	54	cbvralv 2729	. . . . . . . . . . 11
56	52, 55	imbitrrdi 162	. . . . . . . . . 10
57		oveq2 5930	. . . . . . . . . . . . 13
58	57	neeq2d 2386	. . . . . . . . . . . 12
59	58	ralbidv 2497	. . . . . . . . . . 11
60	59	ceqsralv 2794	. . . . . . . . . 10
61	56, 60	sylibrd 169	. . . . . . . . 9
62	61	ancld 325	. . . . . . . 8 $/\$
63		nnleltp1 9385	. . . . . . . . . . . . . 14 $/\$
64		nnz 9345	. . . . . . . . . . . . . . 15
65		nnz 9345	. . . . . . . . . . . . . . 15
66		zleloe 9373	. . . . . . . . . . . . . . 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 2493	. . . . . . . . 9 $/\$
74		r19.26 2623	. . . . . . . . 9 $/\$ $/\$
75	73, 74	bitrdi 196	. . . . . . . 8 $/\$
76	62, 75	sylibrd 169	. . . . . . 7
77	4, 7, 10, 10, 13, 76	nnind 9006	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 8997	. . . . . 6
80	79	ltp1d 8957	. . . . 5
81		breq1 4036	. . . . . . 7
82		df-ne 2368	. . . . . . . . . 10
83	58, 82	bitrdi 196	. . . . . . . . 9
84	83	ralbidv 2497	. . . . . . . 8
85		ralnex 2485	. . . . . . . 8
86	84, 85	bitrdi 196	. . . . . . 7
87	81, 86	imbi12d 234	. . . . . 6
88	87	rspcv 2864	. . . . 5
89	78, 80, 88	mp2d 47	. . . 4
90	89	nrex 2589	. . 3
91		elq 9696	. . . 4
92		rexcom 2661	. . . 4
93	91, 92	bitri 184	. . 3
94	90, 93	mtbir 672	. 2
95	94	nelir 2465	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2167

wne 2367

wnel 2462

wral 2475

wrex 2476 class class class wbr 4033

cfv 5258 (class class class)co 5922

cc 7877

cc0 7879

c1 7880

caddc 7882

clt 8061

cle 8062 # cap 8608

cdiv 8699

cn 8990

c2 9041

cz 9326

cq 9693

crp 9728

csqrt 11161

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-seqfrec 10540 df-exp 10631 df-rsqrt 11163

This theorem is referenced by: sqrt2irr0 12332

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