apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 15692

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 4037	. . 3 # #
2	1	cbvralv 2729	. 2 # #
3		simplll 533	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
4	3	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
5		simplrl 535	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
6	5	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
7		simplrr 536	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
8	7	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
9		simpr 110	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
10		breq2 4037	. . . . . . . . . 10 # #
11		simpllr 534	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 9709	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 411	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9354	. . . . . . . . . . . 12
15		zq 9700	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 9082	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 9717	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2873	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 8055	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 9708	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8633	. . . . . . . . . 10 $/\$ # #
26	22, 24, 25	syl2anc 411	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ # #
27	21, 26	mpbid 147	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ #
28	27	adantr 276	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
29	4, 6, 8, 9, 28	apdifflemf 15690	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 9708	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 9708	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8177	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 5937	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4057	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 15690	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 276	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8337	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11346	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 8055	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8337	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11346	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 8055	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8633	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 411	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 110	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 9715	. . . . . . . 8 $/\$ $\/$
57	5, 7, 56	syl2anc 411	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $\/$
58	55, 57	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $\/$
59	29, 54, 58	mpjaodan 799	. . . . 5 $/\$ # $/\$ $/\$ $/\$ #
60	59	ex 115	. . . 4 $/\$ # $/\$ $/\$ #
61	60	ralrimivva 2579	. . 3 $/\$ # #
62		simpll 527	. . . . 5 $/\$ # $/\$
63		simpr 110	. . . . 5 $/\$ # $/\$
64		simplr 528	. . . . . 6 $/\$ # $/\$ #
65		neg1rr 9096	. . . . . . . 8
66		neg1lt0 9098	. . . . . . . . 9
67		0lt1 8153	. . . . . . . . 9
68		0re 8026	. . . . . . . . . 10
69		1re 8025	. . . . . . . . . 10
70	65, 68, 69	lttri 8131	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 426	. . . . . . . 8
72	65, 71	ltneii 8123	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9358	. . . . . . . 8
75		zq 9700	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9352	. . . . . . . 8
78		zq 9700	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 109	. . . . . . . . . 10 $/\$
81		simpr 110	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2387	. . . . . . . . 9 $/\$
83	80	oveq2d 5938	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5562	. . . . . . . . . 10 $/\$
85	81	oveq2d 5938	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5562	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 4046	. . . . . . . . 9 $/\$ # #
88	82, 87	imbi12d 234	. . . . . . . 8 $/\$ # #
89	88	rspc2gv 2880	. . . . . . 7 $/\$ # #
90	76, 79, 89	syl2anc 411	. . . . . 6 $/\$ # $/\$ # #
91	64, 73, 90	mp2d 47	. . . . 5 $/\$ # $/\$ #
92		simpllr 534	. . . . . 6 $/\$ # $/\$ $/\$ #
93		2cnd 9063	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 528	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 9708	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 9083	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 110	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9337	. . . . . . . . . . . 12
101		zq 9700	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 9713	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 167	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8685	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 9711	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 9708	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 8019	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8633	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 411	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 147	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 9713	. . . . . . . 8 $/\$ #
117	102, 109, 116	syl2anc 411	. . . . . . 7 $/\$ # $/\$ $/\$ #
118	115, 117	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$
119		simpl 109	. . . . . . . . . 10 $/\$
120		simpr 110	. . . . . . . . . 10 $/\$
121	119, 120	neeq12d 2387	. . . . . . . . 9 $/\$
122	119	oveq2d 5938	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5562	. . . . . . . . . 10 $/\$
124	120	oveq2d 5938	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5562	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 4046	. . . . . . . . 9 $/\$ # #
127	121, 126	imbi12d 234	. . . . . . . 8 $/\$ # #
128	127	rspc2gv 2880	. . . . . . 7 $/\$ # #
129	102, 109, 128	syl2anc 411	. . . . . 6 $/\$ # $/\$ $/\$ # #
130	92, 118, 129	mp2d 47	. . . . 5 $/\$ # $/\$ $/\$ #
131	62, 63, 91, 130	apdifflemr 15691	. . . 4 $/\$ # $/\$ #
132	131	ralrimiva 2570	. . 3 $/\$ # #
133	61, 132	impbida 596	. 2 # #
134	2, 133	bitrid 192	1 # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2167

wne 2367

wral 2475 class class class wbr 4033

cfv 5258 (class class class)co 5922

cc 7877

cr 7878

cc0 7879

c1 7880

caddc 7882

cmul 7884

clt 8061

cmin 8197

cneg 8198 # cap 8608

cdiv 8699

c2 9041

cz 9326

cq 9693

cabs 11162

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-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164

This theorem is referenced by: (None)

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