apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 14080

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 3993	. . 3 # #
2	1	cbvralv 2696	. 2 # #
3		simplll 528	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
4	3	adantr 274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
5		simplrl 530	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
6	5	adantr 274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
7		simplrr 531	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
8	7	adantr 274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
9		simpr 109	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
10		breq2 3993	. . . . . . . . . 10 # #
11		simpllr 529	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 9594	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 409	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9240	. . . . . . . . . . . 12
15		zq 9585	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 8970	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 9602	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1233	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2839	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 7948	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 9593	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8525	. . . . . . . . . 10 $/\$ # #
26	22, 24, 25	syl2anc 409	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ # #
27	21, 26	mpbid 146	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ #
28	27	adantr 274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
29	4, 6, 8, 9, 28	apdifflemf 14078	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 9593	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 9593	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8070	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 5868	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4013	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 14078	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 274	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8230	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11145	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 7948	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8230	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11145	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 7948	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8525	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 409	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 146	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 109	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 9600	. . . . . . . 8 $/\$ $\/$
57	5, 7, 56	syl2anc 409	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $\/$
58	55, 57	mpbid 146	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $\/$
59	29, 54, 58	mpjaodan 793	. . . . 5 $/\$ # $/\$ $/\$ $/\$ #
60	59	ex 114	. . . 4 $/\$ # $/\$ $/\$ #
61	60	ralrimivva 2552	. . 3 $/\$ # #
62		simpll 524	. . . . 5 $/\$ # $/\$
63		simpr 109	. . . . 5 $/\$ # $/\$
64		simplr 525	. . . . . 6 $/\$ # $/\$ #
65		neg1rr 8984	. . . . . . . 8
66		neg1lt0 8986	. . . . . . . . 9
67		0lt1 8046	. . . . . . . . 9
68		0re 7920	. . . . . . . . . 10
69		1re 7919	. . . . . . . . . 10
70	65, 68, 69	lttri 8024	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 424	. . . . . . . 8
72	65, 71	ltneii 8016	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9244	. . . . . . . 8
75		zq 9585	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9238	. . . . . . . 8
78		zq 9585	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 108	. . . . . . . . . 10 $/\$
81		simpr 109	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2360	. . . . . . . . 9 $/\$
83	80	oveq2d 5869	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5500	. . . . . . . . . 10 $/\$
85	81	oveq2d 5869	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5500	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 4002	. . . . . . . . 9 $/\$ # #
88	82, 87	imbi12d 233	. . . . . . . 8 $/\$ # #
89	88	rspc2gv 2846	. . . . . . 7 $/\$ # #
90	76, 79, 89	syl2anc 409	. . . . . 6 $/\$ # $/\$ # #
91	64, 73, 90	mp2d 47	. . . . 5 $/\$ # $/\$ #
92		simpllr 529	. . . . . 6 $/\$ # $/\$ $/\$ #
93		2cnd 8951	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 525	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 9593	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 8971	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 109	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9223	. . . . . . . . . . . 12
101		zq 9585	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 9598	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 166	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8576	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 9596	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 9593	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 7913	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8525	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 409	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 146	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 9598	. . . . . . . 8 $/\$ #
117	102, 109, 116	syl2anc 409	. . . . . . 7 $/\$ # $/\$ $/\$ #
118	115, 117	mpbid 146	. . . . . 6 $/\$ # $/\$ $/\$
119		simpl 108	. . . . . . . . . 10 $/\$
120		simpr 109	. . . . . . . . . 10 $/\$
121	119, 120	neeq12d 2360	. . . . . . . . 9 $/\$
122	119	oveq2d 5869	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5500	. . . . . . . . . 10 $/\$
124	120	oveq2d 5869	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5500	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 4002	. . . . . . . . 9 $/\$ # #
127	121, 126	imbi12d 233	. . . . . . . 8 $/\$ # #
128	127	rspc2gv 2846	. . . . . . 7 $/\$ # #
129	102, 109, 128	syl2anc 409	. . . . . 6 $/\$ # $/\$ $/\$ # #
130	92, 118, 129	mp2d 47	. . . . 5 $/\$ # $/\$ $/\$ #
131	62, 63, 91, 130	apdifflemr 14079	. . . 4 $/\$ # $/\$ #
132	131	ralrimiva 2543	. . 3 $/\$ # #
133	61, 132	impbida 591	. 2 # #
134	2, 133	syl5bb 191	1 # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141

wne 2340

wral 2448 class class class wbr 3989

cfv 5198 (class class class)co 5853

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cmin 8090

cneg 8091 # cap 8500

cdiv 8589

c2 8929

cz 9212

cq 9578

cabs 10961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963

This theorem is referenced by: (None)

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