apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 14045

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 3991	. . 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 3991	. . . . . . . . . 10 # #
11		simpllr 529	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 9587	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 409	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9233	. . . . . . . . . . . 12
15		zq 9578	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 8963	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 9595	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1233	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2839	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 7941	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 9586	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8518	. . . . . . . . . 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 14043	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 9586	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 9586	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 8063	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 5866	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 4011	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 14043	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 274	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8223	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 11138	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 7941	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8223	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 11138	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 7941	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8518	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 409	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 146	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 109	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 9593	. . . . . . . 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 8977	. . . . . . . 8
66		neg1lt0 8979	. . . . . . . . 9
67		0lt1 8039	. . . . . . . . 9
68		0re 7913	. . . . . . . . . 10
69		1re 7912	. . . . . . . . . 10
70	65, 68, 69	lttri 8017	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 424	. . . . . . . 8
72	65, 71	ltneii 8009	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9237	. . . . . . . 8
75		zq 9578	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9231	. . . . . . . 8
78		zq 9578	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 108	. . . . . . . . . 10 $/\$
81		simpr 109	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2360	. . . . . . . . 9 $/\$
83	80	oveq2d 5867	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5498	. . . . . . . . . 10 $/\$
85	81	oveq2d 5867	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5498	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 4000	. . . . . . . . 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 8944	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 525	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 9586	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 8964	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 109	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9216	. . . . . . . . . . . 12
101		zq 9578	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 9591	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 166	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8569	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 9589	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 9586	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 7906	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8518	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 409	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 146	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 9591	. . . . . . . 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 5867	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5498	. . . . . . . . . 10 $/\$
124	120	oveq2d 5867	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5498	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 4000	. . . . . . . . 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 14044	. . . 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 3987

cfv 5196 (class class class)co 5851

cc 7765

cr 7766

cc0 7767

c1 7768

caddc 7770

cmul 7772

clt 7947

cmin 8083

cneg 8084 # cap 8493

cdiv 8582

c2 8922

cz 9205

cq 9571

cabs 10954

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 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887

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 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-seqfrec 10395 df-exp 10469 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956

This theorem is referenced by: (None)

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