apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 17109

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 4134	. . 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 4134	. . . . . . . . . 10 # #
11		simpllr 540	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 10037	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 415	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9674	. . . . . . . . . . . 12
15		zq 10028	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 9397	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 10045	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1278	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2934	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 8354	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 10036	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8935	. . . . . . . . . 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 17107	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 10036	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 10036	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8477	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 6100	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4154	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 17107	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 276	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8637	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11949	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 8354	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8637	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11949	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 8354	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8935	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 415	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 110	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 10043	. . . . . . . 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 9411	. . . . . . . 8
66		neg1lt0 9413	. . . . . . . . 9
67		0lt1 8453	. . . . . . . . 9
68		0re 8326	. . . . . . . . . 10
69		1re 8325	. . . . . . . . . 10
70	65, 68, 69	lttri 8430	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 430	. . . . . . . 8
72	65, 71	ltneii 8422	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9678	. . . . . . . 8
75		zq 10028	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9672	. . . . . . . 8
78		zq 10028	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 109	. . . . . . . . . 10 $/\$
81		simpr 110	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2440	. . . . . . . . 9 $/\$
83	80	oveq2d 6101	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5699	. . . . . . . . . 10 $/\$
85	81	oveq2d 6101	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5699	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 4143	. . . . . . . . 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 9378	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 533	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 10036	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 9398	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 110	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9657	. . . . . . . . . . . 12
101		zq 10028	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 10041	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 415	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 167	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8987	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 10039	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 415	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 10036	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 8319	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8935	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 415	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 147	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 10041	. . . . . . . 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 6101	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5699	. . . . . . . . . 10 $/\$
124	120	oveq2d 6101	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5699	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 4143	. . . . . . . . 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 17108	. . . 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 4130

cfv 5377 (class class class)co 6085

cc 8177

cr 8178

cc0 8179

c1 8180

caddc 8182

cmul 8184

clt 8360

cmin 8497

cneg 8498 # cap 8910

cdiv 9003

c2 9356

cz 9646

cq 10021

cabs 11765

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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

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 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8904 df-ap 8911 df-div 9004 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-n0 9566 df-z 9647 df-uz 9924 df-q 10022 df-rp 10057 df-seqfrec 10887 df-exp 10978 df-cj 11609 df-re 11610 df-im 11611 df-rsqrt 11766 df-abs 11767

This theorem is used by: (None)

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