apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 15608

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 4034	. . 3 # #
2	1	cbvralv 2726	. 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 4034	. . . . . . . . . 10 # #
11		simpllr 534	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 9703	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 411	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9348	. . . . . . . . . . . 12
15		zq 9694	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 9076	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 9711	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1249	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2870	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 8050	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 9702	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8627	. . . . . . . . . 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 15606	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 9702	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 9702	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8172	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 5934	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4054	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 15606	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 276	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8332	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11328	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 8050	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8332	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11328	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 8050	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8627	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 411	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 110	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 9709	. . . . . . . 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 2576	. . 3 $/\$ # #
62		simpll 527	. . . . 5 $/\$ # $/\$
63		simpr 110	. . . . 5 $/\$ # $/\$
64		simplr 528	. . . . . 6 $/\$ # $/\$ #
65		neg1rr 9090	. . . . . . . 8
66		neg1lt0 9092	. . . . . . . . 9
67		0lt1 8148	. . . . . . . . 9
68		0re 8021	. . . . . . . . . 10
69		1re 8020	. . . . . . . . . 10
70	65, 68, 69	lttri 8126	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 426	. . . . . . . 8
72	65, 71	ltneii 8118	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9352	. . . . . . . 8
75		zq 9694	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9346	. . . . . . . 8
78		zq 9694	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 109	. . . . . . . . . 10 $/\$
81		simpr 110	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2384	. . . . . . . . 9 $/\$
83	80	oveq2d 5935	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5559	. . . . . . . . . 10 $/\$
85	81	oveq2d 5935	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5559	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 4043	. . . . . . . . 9 $/\$ # #
88	82, 87	imbi12d 234	. . . . . . . 8 $/\$ # #
89	88	rspc2gv 2877	. . . . . . 7 $/\$ # #
90	76, 79, 89	syl2anc 411	. . . . . 6 $/\$ # $/\$ # #
91	64, 73, 90	mp2d 47	. . . . 5 $/\$ # $/\$ #
92		simpllr 534	. . . . . 6 $/\$ # $/\$ $/\$ #
93		2cnd 9057	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 528	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 9702	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 9077	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 110	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9331	. . . . . . . . . . . 12
101		zq 9694	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 9707	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 167	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8679	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 9705	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 9702	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 8014	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8627	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 411	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 147	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 9707	. . . . . . . 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 2384	. . . . . . . . 9 $/\$
122	119	oveq2d 5935	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5559	. . . . . . . . . 10 $/\$
124	120	oveq2d 5935	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5559	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 4043	. . . . . . . . 9 $/\$ # #
127	121, 126	imbi12d 234	. . . . . . . 8 $/\$ # #
128	127	rspc2gv 2877	. . . . . . 7 $/\$ # #
129	102, 109, 128	syl2anc 411	. . . . . 6 $/\$ # $/\$ $/\$ # #
130	92, 118, 129	mp2d 47	. . . . 5 $/\$ # $/\$ $/\$ #
131	62, 63, 91, 130	apdifflemr 15607	. . . 4 $/\$ # $/\$ #
132	131	ralrimiva 2567	. . 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 2164

wne 2364

wral 2472 class class class wbr 4030

cfv 5255 (class class class)co 5919

cc 7872

cr 7873

cc0 7874

c1 7875

caddc 7877

cmul 7879

clt 8056

cmin 8192

cneg 8193 # cap 8602

cdiv 8693

c2 9035

cz 9320

cq 9687

cabs 11144

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-seqfrec 10522 df-exp 10613 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146

This theorem is referenced by: (None)

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