apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 14452

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 4004	. . 3 # #
2	1	cbvralv 2703	. 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 4004	. . . . . . . . . 10 # #
11		simpllr 534	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 9624	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 411	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9270	. . . . . . . . . . . 12
15		zq 9615	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 9000	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 9632	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1238	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2846	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 7976	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 9623	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8553	. . . . . . . . . 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 14450	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 110	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 9623	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 9623	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8098	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 5884	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4024	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 276	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 14450	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 276	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8258	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11174	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 7976	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8258	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11174	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 7976	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8553	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 411	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 110	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 9630	. . . . . . . 8 $/\$ $\/$
57	5, 7, 56	syl2anc 411	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $\/$
58	55, 57	mpbid 147	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $\/$
59	29, 54, 58	mpjaodan 798	. . . . 5 $/\$ # $/\$ $/\$ $/\$ #
60	59	ex 115	. . . 4 $/\$ # $/\$ $/\$ #
61	60	ralrimivva 2559	. . 3 $/\$ # #
62		simpll 527	. . . . 5 $/\$ # $/\$
63		simpr 110	. . . . 5 $/\$ # $/\$
64		simplr 528	. . . . . 6 $/\$ # $/\$ #
65		neg1rr 9014	. . . . . . . 8
66		neg1lt0 9016	. . . . . . . . 9
67		0lt1 8074	. . . . . . . . 9
68		0re 7948	. . . . . . . . . 10
69		1re 7947	. . . . . . . . . 10
70	65, 68, 69	lttri 8052	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 426	. . . . . . . 8
72	65, 71	ltneii 8044	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9274	. . . . . . . 8
75		zq 9615	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9268	. . . . . . . 8
78		zq 9615	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 109	. . . . . . . . . 10 $/\$
81		simpr 110	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2367	. . . . . . . . 9 $/\$
83	80	oveq2d 5885	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5515	. . . . . . . . . 10 $/\$
85	81	oveq2d 5885	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5515	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 4013	. . . . . . . . 9 $/\$ # #
88	82, 87	imbi12d 234	. . . . . . . 8 $/\$ # #
89	88	rspc2gv 2853	. . . . . . 7 $/\$ # #
90	76, 79, 89	syl2anc 411	. . . . . 6 $/\$ # $/\$ # #
91	64, 73, 90	mp2d 47	. . . . 5 $/\$ # $/\$ #
92		simpllr 534	. . . . . 6 $/\$ # $/\$ $/\$ #
93		2cnd 8981	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 528	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 9623	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 9001	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 110	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9253	. . . . . . . . . . . 12
101		zq 9615	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 9628	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 167	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8604	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 9626	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 411	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 9623	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 7941	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8553	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 411	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 147	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 9628	. . . . . . . 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 2367	. . . . . . . . 9 $/\$
122	119	oveq2d 5885	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5515	. . . . . . . . . 10 $/\$
124	120	oveq2d 5885	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5515	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 4013	. . . . . . . . 9 $/\$ # #
127	121, 126	imbi12d 234	. . . . . . . 8 $/\$ # #
128	127	rspc2gv 2853	. . . . . . 7 $/\$ # #
129	102, 109, 128	syl2anc 411	. . . . . 6 $/\$ # $/\$ $/\$ # #
130	92, 118, 129	mp2d 47	. . . . 5 $/\$ # $/\$ $/\$ #
131	62, 63, 91, 130	apdifflemr 14451	. . . 4 $/\$ # $/\$ #
132	131	ralrimiva 2550	. . 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 708

wceq 1353

wcel 2148

wne 2347

wral 2455 class class class wbr 4000

cfv 5212 (class class class)co 5869

cc 7800

cr 7801

cc0 7802

c1 7803

caddc 7805

cmul 7807

clt 7982

cmin 8118

cneg 8119 # cap 8528

cdiv 8618

c2 8959

cz 9242

cq 9608

cabs 10990

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992

This theorem is referenced by: (None)

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