apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 13927

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 3986	. . 3 # #
2	1	cbvralv 2692	. 2 # #
3		simplll 523	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
4	3	adantr 274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
5		simplrl 525	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
6	5	adantr 274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
7		simplrr 526	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$
8	7	adantr 274	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
9		simpr 109	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$
10		breq2 3986	. . . . . . . . . 10 # #
11		simpllr 524	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 9573	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 409	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9219	. . . . . . . . . . . 12
15		zq 9564	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 8949	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 9581	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1228	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2835	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 7927	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 9572	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8504	. . . . . . . . . 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 13925	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 9572	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 9572	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8049	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 5857	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4006	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 13925	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 274	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8209	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11123	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 7927	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8209	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11123	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 7927	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8504	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 409	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 146	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 109	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 9579	. . . . . . . 8 $/\$ $\/$
57	5, 7, 56	syl2anc 409	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $\/$
58	55, 57	mpbid 146	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $\/$
59	29, 54, 58	mpjaodan 788	. . . . 5 $/\$ # $/\$ $/\$ $/\$ #
60	59	ex 114	. . . 4 $/\$ # $/\$ $/\$ #
61	60	ralrimivva 2548	. . 3 $/\$ # #
62		simpll 519	. . . . 5 $/\$ # $/\$
63		simpr 109	. . . . 5 $/\$ # $/\$
64		simplr 520	. . . . . 6 $/\$ # $/\$ #
65		neg1rr 8963	. . . . . . . 8
66		neg1lt0 8965	. . . . . . . . 9
67		0lt1 8025	. . . . . . . . 9
68		0re 7899	. . . . . . . . . 10
69		1re 7898	. . . . . . . . . 10
70	65, 68, 69	lttri 8003	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 423	. . . . . . . 8
72	65, 71	ltneii 7995	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9223	. . . . . . . 8
75		zq 9564	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9217	. . . . . . . 8
78		zq 9564	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 108	. . . . . . . . . 10 $/\$
81		simpr 109	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2356	. . . . . . . . 9 $/\$
83	80	oveq2d 5858	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5490	. . . . . . . . . 10 $/\$
85	81	oveq2d 5858	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5490	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 3995	. . . . . . . . 9 $/\$ # #
88	82, 87	imbi12d 233	. . . . . . . 8 $/\$ # #
89	88	rspc2gv 2842	. . . . . . 7 $/\$ # #
90	76, 79, 89	syl2anc 409	. . . . . 6 $/\$ # $/\$ # #
91	64, 73, 90	mp2d 47	. . . . 5 $/\$ # $/\$ #
92		simpllr 524	. . . . . 6 $/\$ # $/\$ $/\$ #
93		2cnd 8930	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 520	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 9572	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 8950	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 109	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9202	. . . . . . . . . . . 12
101		zq 9564	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 9577	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 166	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8555	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 9575	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 9572	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 7892	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8504	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 409	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 146	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 9577	. . . . . . . 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 2356	. . . . . . . . 9 $/\$
122	119	oveq2d 5858	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5490	. . . . . . . . . 10 $/\$
124	120	oveq2d 5858	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5490	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 3995	. . . . . . . . 9 $/\$ # #
127	121, 126	imbi12d 233	. . . . . . . 8 $/\$ # #
128	127	rspc2gv 2842	. . . . . . 7 $/\$ # #
129	102, 109, 128	syl2anc 409	. . . . . 6 $/\$ # $/\$ $/\$ # #
130	92, 118, 129	mp2d 47	. . . . 5 $/\$ # $/\$ $/\$ #
131	62, 63, 91, 130	apdifflemr 13926	. . . 4 $/\$ # $/\$ #
132	131	ralrimiva 2539	. . 3 $/\$ # #
133	61, 132	impbida 586	. 2 # #
134	2, 133	syl5bb 191	1 # #

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1343

wcel 2136

wne 2336

wral 2444 class class class wbr 3982

cfv 5188 (class class class)co 5842

cc 7751

cr 7752

cc0 7753

c1 7754

caddc 7756

cmul 7758

clt 7933

cmin 8069

cneg 8070 # cap 8479

cdiv 8568

c2 8908

cz 9191

cq 9557

cabs 10939

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941

This theorem is referenced by: (None)

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