Theorem sqrt2irrap 12873

Description: The square root of 2 is irrational. That is, for any rational number, ( sqr ` 2 ) is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 12855. (Contributed by Jim Kingdon, 2-Oct-2021.)

Assertion

Ref	Expression
sqrt2irrap	|- ( Q e. QQ -> ( sqr ` 2 ) # Q )

Proof of Theorem sqrt2irrap

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elq 9953	. . 3 |- ( Q e. QQ <-> E. a e. ZZ E. b e. NN Q = ( a / b ) )
2	1	biimpi 120	. 2 |- ( Q e. QQ -> E. a e. ZZ E. b e. NN Q = ( a / b ) )
3		simplrl 537	. . . . . . . . 9 |- ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) -> a e. ZZ )
4	3	adantr 276	. . . . . . . 8 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> a e. ZZ )
5		simplrr 538	. . . . . . . . 9 |- ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) -> b e. NN )
6	5	adantr 276	. . . . . . . 8 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> b e. NN )
7		znq 9955	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. NN ) -> ( a / b ) e. QQ )
8		qre 9956	. . . . . . . . 9 |- ( ( a / b ) e. QQ -> ( a / b ) e. RR )
9	7, 8	syl 14	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. NN ) -> ( a / b ) e. RR )
10	4, 6, 9	syl2anc 411	. . . . . . 7 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( a / b ) e. RR )
11		sqrt2re 12856	. . . . . . . 8 |- ( sqr ` 2 ) e. RR
12	11	a1i 9	. . . . . . 7 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( sqr ` 2 ) e. RR )
13		0red 8274	. . . . . . . 8 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> 0 e. RR )
14	4	zcnd 9700	. . . . . . . . . 10 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> a e. CC )
15	6	nncnd 9250	. . . . . . . . . 10 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> b e. CC )
16	6	nnap0d 9282	. . . . . . . . . 10 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> b # 0 )
17	14, 15, 16	divrecapd 9066	. . . . . . . . 9 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( a / b ) = ( a x. ( 1 / b ) ) )
18	4	zred 9699	. . . . . . . . . 10 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> a e. RR )
19	6	nnrecred 9283	. . . . . . . . . 10 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( 1 / b ) e. RR )
20		simpr 110	. . . . . . . . . 10 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> a <_ 0 )
21		1red 8288	. . . . . . . . . . 11 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> 1 e. RR )
22	6	nnrpd 10026	. . . . . . . . . . 11 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> b e. RR+ )
23		0le1 8754	. . . . . . . . . . . 12 |- 0 <_ 1
24	23	a1i 9	. . . . . . . . . . 11 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> 0 <_ 1 )
25	21, 22, 24	divge0d 10069	. . . . . . . . . 10 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> 0 <_ ( 1 / b ) )
26		mulle0r 9217	. . . . . . . . . 10 |- ( ( ( a e. RR /\ ( 1 / b ) e. RR ) /\ ( a <_ 0 /\ 0 <_ ( 1 / b ) ) ) -> ( a x. ( 1 / b ) ) <_ 0 )
27	18, 19, 20, 25, 26	syl22anc 1275	. . . . . . . . 9 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( a x. ( 1 / b ) ) <_ 0 )
28	17, 27	eqbrtrd 4130	. . . . . . . 8 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( a / b ) <_ 0 )
29		2re 9306	. . . . . . . . . 10 |- 2 e. RR
30		2pos 9327	. . . . . . . . . 10 |- 0 < 2
31	29, 30	sqrtgt0ii 11812	. . . . . . . . 9 |- 0 < ( sqr ` 2 )
32	31	a1i 9	. . . . . . . 8 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> 0 < ( sqr ` 2 ) )
33	10, 13, 12, 28, 32	lelttrd 8397	. . . . . . 7 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( a / b ) < ( sqr ` 2 ) )
34	10, 12, 33	gtapd 8910	. . . . . 6 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ a <_ 0 ) -> ( sqr ` 2 ) # ( a / b ) )
35	3	adantr 276	. . . . . . . 8 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ 0 < a ) -> a e. ZZ )
36		simpr 110	. . . . . . . 8 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ 0 < a ) -> 0 < a )
37		elnnz 9586	. . . . . . . 8 |- ( a e. NN <-> ( a e. ZZ /\ 0 < a ) )
38	35, 36, 37	sylanbrc 417	. . . . . . 7 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ 0 < a ) -> a e. NN )
39	5	adantr 276	. . . . . . 7 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ 0 < a ) -> b e. NN )
40		sqrt2irraplemnn 12872	. . . . . . 7 |- ( ( a e. NN /\ b e. NN ) -> ( sqr ` 2 ) # ( a / b ) )
41	38, 39, 40	syl2anc 411	. . . . . 6 |- ( ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) /\ 0 < a ) -> ( sqr ` 2 ) # ( a / b ) )
42		0z 9587	. . . . . . . . 9 |- 0 e. ZZ
43		zlelttric 9621	. . . . . . . . 9 |- ( ( a e. ZZ /\ 0 e. ZZ ) -> ( a <_ 0 \/ 0 < a ) )
44	42, 43	mpan2 425	. . . . . . . 8 |- ( a e. ZZ -> ( a <_ 0 \/ 0 < a ) )
45	44	ad2antrl 490	. . . . . . 7 |- ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) -> ( a <_ 0 \/ 0 < a ) )
46	45	adantr 276	. . . . . 6 |- ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) -> ( a <_ 0 \/ 0 < a ) )
47	34, 41, 46	mpjaodan 806	. . . . 5 |- ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) -> ( sqr ` 2 ) # ( a / b ) )
48		simpr 110	. . . . 5 |- ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) -> Q = ( a / b ) )
49	47, 48	breqtrrd 4136	. . . 4 |- ( ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) /\ Q = ( a / b ) ) -> ( sqr ` 2 ) # Q )
50	49	ex 115	. . 3 |- ( ( Q e. QQ /\ ( a e. ZZ /\ b e. NN ) ) -> ( Q = ( a / b ) -> ( sqr ` 2 ) # Q ) )
51	50	rexlimdvva 2668	. 2 |- ( Q e. QQ -> ( E. a e. ZZ E. b e. NN Q = ( a / b ) -> ( sqr ` 2 ) # Q ) )
52	2, 51	mpd 13	1 |- ( Q e. QQ -> ( sqr ` 2 ) # Q )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   RRcr 8125   0cc0 8126   1c1 8127   x. cmul 8131   < clt 8307   <_ cle 8308   # cap 8854   / cdiv 8945   NNcn 9236   2c2 9287   ZZcz 9576   QQcq 9950   sqrcsqrt 11677

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-sup 7274  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-dvds 12470  df-gcd 12646  df-prm 12801

This theorem is referenced by:  2irrexpqap  15835