sqrt2irr - Intuitionistic Logic Explorer

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Theorem sqrt2irr 12355

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 12373 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12354, 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 9019	. . . . . 6
2		breq2 4038	. . . . . . . . 9
3	2	imbi1d 231	. . . . . . . 8
4	3	ralbidv 2497	. . . . . . 7
5		breq2 4038	. . . . . . . . 9
6	5	imbi1d 231	. . . . . . . 8
7	6	ralbidv 2497	. . . . . . 7
8		breq2 4038	. . . . . . . . 9
9	8	imbi1d 231	. . . . . . . 8
10	9	ralbidv 2497	. . . . . . 7
11		nnnlt1 9033	. . . . . . . . 9
12	11	pm2.21d 620	. . . . . . . 8
13	12	rgen 2550	. . . . . . 7
14		nnrp 9755	. . . . . . . . . . . . . 14
15		rphalflt 9775	. . . . . . . . . . . . . 14
16	14, 15	syl 14	. . . . . . . . . . . . 13
17		breq1 4037	. . . . . . . . . . . . . . . 16
18		oveq2 5933	. . . . . . . . . . . . . . . . . 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 9348	. . . . . . . . . . . . . . . . . . 19
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
28		nncn 9015	. . . . . . . . . . . . . . . . . . 19
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
30		2cnd 9080	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
31		nnap0 9036	. . . . . . . . . . . . . . . . . . 19 #
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
33		2ap0 9100	. . . . . . . . . . . . . . . . . . 19 #
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ #
35	27, 29, 30, 32, 34	divcanap7d 8863	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
36	25, 35	eqtr4d 2232	. . . . . . . . . . . . . . . 16 $/\$ $/\$
37		simplr 528	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
38		simpll 527	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
39	37, 38, 25	sqrt2irrlem 12354	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
42		oveq1 5932	. . . . . . . . . . . . . . . . . . . . 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 5932	. . . . . . . . . . . . 13
54	53	neeq2d 2386	. . . . . . . . . . . 12
55	54	cbvralv 2729	. . . . . . . . . . 11
56	52, 55	imbitrrdi 162	. . . . . . . . . 10
57		oveq2 5933	. . . . . . . . . . . . 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 9402	. . . . . . . . . . . . . 14 $/\$
64		nnz 9362	. . . . . . . . . . . . . . 15
65		nnz 9362	. . . . . . . . . . . . . . 15
66		zleloe 9390	. . . . . . . . . . . . . . 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 9023	. . . . . 6
78	1, 77	syl 14	. . . . 5
79		nnre 9014	. . . . . 6
80	79	ltp1d 8974	. . . . 5
81		breq1 4037	. . . . . . 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 9713	. . . 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 4034

cfv 5259 (class class class)co 5925

cc 7894

cc0 7896

c1 7897

caddc 7899

clt 8078

cle 8079 # cap 8625

cdiv 8716

cn 9007

c2 9058

cz 9343

cq 9710

crp 9745

csqrt 11178

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 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

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 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-rp 9746 df-seqfrec 10557 df-exp 10648 df-rsqrt 11180

This theorem is referenced by: sqrt2irr0 12357

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