apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 17316

Description: The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.)

**Assertion**
Ref	Expression
apdiff	# #

Distinct variable group:

Proof of Theorem apdiff Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4135	. . 3 # #
2	1	cbvralv 2786	. 2 # #
3		simplll 539	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
4	3	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
5		simplrl 541	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
6	5	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
7		simplrr 542	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
8	7	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
9		simpr 110	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
10		breq2 4135	. . . . . . . . . 10 # #
11		simpllr 540	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 10052	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 415	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9684	. . . . . . . . . . . 12
15		zq 10043	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 9406	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 10060	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1278	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2934	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 8361	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 10051	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8943	. . . . . . . . . 10 $/\$ # #
26	22, 24, 25	syl2anc 415	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ # #
27	21, 26	mpbid 147	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ #
28	27	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
29	4, 6, 8, 9, 28	apdifflemf 17314	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 10051	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 10051	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8485	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 6103	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4155	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 17314	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 276	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8645	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11971	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 8361	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8645	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11971	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 8361	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8943	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 415	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 110	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 10058	. . . . . . . 8 $/\$ $\/$
57	5, 7, 56	syl2anc 415	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $\/$
58	55, 57	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $\/$
59	29, 54, 58	mpjaodan 810	. . . . 5 $/\$ # $/\$ $/\$ $/\$ #
60	59	ex 115	. . . 4 $/\$ # $/\$ $/\$ #
61	60	ralrimivva 2632	. . 3 $/\$ # #
62		simpll 531	. . . . 5 $/\$ # $/\$
63		simpr 110	. . . . 5 $/\$ # $/\$
64		simplr 533	. . . . . 6 $/\$ # $/\$ #
65		neg1rr 9420	. . . . . . . 8
66		neg1lt0 9422	. . . . . . . . 9
67		0lt1 8461	. . . . . . . . 9
68		0re 8333	. . . . . . . . . 10
69		1re 8332	. . . . . . . . . 10
70	65, 68, 69	lttri 8438	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 430	. . . . . . . 8
72	65, 71	ltneii 8430	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9688	. . . . . . . 8
75		zq 10043	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9682	. . . . . . . 8
78		zq 10043	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 109	. . . . . . . . . 10 $/\$
81		simpr 110	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2440	. . . . . . . . 9 $/\$
83	80	oveq2d 6104	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5702	. . . . . . . . . 10 $/\$
85	81	oveq2d 6104	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5702	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 4144	. . . . . . . . 9 $/\$ # #
88	82, 87	imbi12d 234	. . . . . . . 8 $/\$ # #
89	88	rspc2gv 2942	. . . . . . 7 $/\$ # #
90	76, 79, 89	syl2anc 415	. . . . . 6 $/\$ # $/\$ # #
91	64, 73, 90	mp2d 47	. . . . 5 $/\$ # $/\$ #
92		simpllr 540	. . . . . 6 $/\$ # $/\$ $/\$ #
93		2cnd 9387	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 533	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 10051	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 9407	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 110	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9667	. . . . . . . . . . . 12
101		zq 10043	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 10056	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 415	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 167	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8995	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 10054	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 415	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 10051	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 8326	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8943	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 415	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 147	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 10056	. . . . . . . 8 $/\$ #
117	102, 109, 116	syl2anc 415	. . . . . . 7 $/\$ # $/\$ $/\$ #
118	115, 117	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$
119		simpl 109	. . . . . . . . . 10 $/\$
120		simpr 110	. . . . . . . . . 10 $/\$
121	119, 120	neeq12d 2440	. . . . . . . . 9 $/\$
122	119	oveq2d 6104	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5702	. . . . . . . . . 10 $/\$
124	120	oveq2d 6104	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5702	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 4144	. . . . . . . . 9 $/\$ # #
127	121, 126	imbi12d 234	. . . . . . . 8 $/\$ # #
128	127	rspc2gv 2942	. . . . . . 7 $/\$ # #
129	102, 109, 128	syl2anc 415	. . . . . 6 $/\$ # $/\$ $/\$ # #
130	92, 118, 129	mp2d 47	. . . . 5 $/\$ # $/\$ $/\$ #
131	62, 63, 91, 130	apdifflemr 17315	. . . 4 $/\$ # $/\$ #
132	131	ralrimiva 2623	. . 3 $/\$ # #
133	61, 132	impbida 604	. 2 # #
134	2, 133	bitrid 192	1 # #

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209

wne 2420

wral 2528 class class class wbr 4131

cfv 5380 (class class class)co 6088

cc 8184

cr 8185

cc0 8186

c1 8187

caddc 8189

cmul 8191

clt 8367

cmin 8505

cneg 8506 # cap 8918

cdiv 9011

c2 9365

cz 9656

cq 10036

cabs 11786

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-n0 9576 df-z 9657 df-uz 9939 df-q 10037 df-rp 10073 df-seqfrec 10907 df-exp 10998 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788

This theorem is used by: (None)

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