apdiff - Mathbox for Jim Kingdon

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Theorem apdiff 13416

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 3941	. . 3 # #
2	1	cbvralv 2657	. 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 3941	. . . . . . . . . 10 # #
11		simpllr 524	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ #
12		qaddcl 9454	. . . . . . . . . . . 12 $/\$
13	5, 7, 12	syl2anc 409	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
14		2z 9106	. . . . . . . . . . . 12
15		zq 9445	. . . . . . . . . . . 12
16	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
17		2ne0 8836	. . . . . . . . . . . 12
18	17	a1i 9	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
19		qdivcl 9462	. . . . . . . . . . 11 $/\$ $/\$
20	13, 16, 18, 19	syl3anc 1217	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
21	10, 11, 20	rspcdva 2798	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
22	3	recnd 7818	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
23		qcn 9453	. . . . . . . . . . 11
24	20, 23	syl 14	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
25		apsym 8392	. . . . . . . . . 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 13414	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
30	3	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
31	7	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
32	5	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
33		simpr 109	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
34		qcn 9453	. . . . . . . . . . . . 13
35	5, 34	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
36		qcn 9453	. . . . . . . . . . . . 13
37	7, 36	syl 14	. . . . . . . . . . . 12 $/\$ # $/\$ $/\$ $/\$
38	35, 37	addcomd 7937	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$
39	38	oveq1d 5797	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$
40	39, 27	eqbrtrrd 3960	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ #
41	40	adantr 274	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$ #
42	30, 31, 32, 33, 41	apdifflemf 13414	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ #
43	22	adantr 274	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
44	31, 36	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	subcld 8097	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
46	45	abscld 10985	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
47	46	recnd 7818	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
48	32, 34	syl 14	. . . . . . . . . . 11 $/\$ # $/\$ $/\$ $/\$ $/\$
49	43, 48	subcld 8097	. . . . . . . . . 10 $/\$ # $/\$ $/\$ $/\$ $/\$
50	49	abscld 10985	. . . . . . . . 9 $/\$ # $/\$ $/\$ $/\$ $/\$
51	50	recnd 7818	. . . . . . . 8 $/\$ # $/\$ $/\$ $/\$ $/\$
52		apsym 8392	. . . . . . . 8 $/\$ # #
53	47, 51, 52	syl2anc 409	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$ $/\$ # #
54	42, 53	mpbid 146	. . . . . 6 $/\$ # $/\$ $/\$ $/\$ $/\$ #
55		simpr 109	. . . . . . 7 $/\$ # $/\$ $/\$ $/\$
56		qlttri2 9460	. . . . . . . 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 2517	. . 3 $/\$ # #
62		simpll 519	. . . . 5 $/\$ # $/\$
63		simpr 109	. . . . 5 $/\$ # $/\$
64		simplr 520	. . . . . 6 $/\$ # $/\$ #
65		neg1rr 8850	. . . . . . . 8
66		neg1lt0 8852	. . . . . . . . 9
67		0lt1 7913	. . . . . . . . 9
68		0re 7790	. . . . . . . . . 10
69		1re 7789	. . . . . . . . . 10
70	65, 68, 69	lttri 7892	. . . . . . . . 9 $/\$
71	66, 67, 70	mp2an 423	. . . . . . . 8
72	65, 71	ltneii 7884	. . . . . . 7
73	72	a1i 9	. . . . . 6 $/\$ # $/\$
74		neg1z 9110	. . . . . . . 8
75		zq 9445	. . . . . . . 8
76	74, 75	mp1i 10	. . . . . . 7 $/\$ # $/\$
77		1z 9104	. . . . . . . 8
78		zq 9445	. . . . . . . 8
79	77, 78	mp1i 10	. . . . . . 7 $/\$ # $/\$
80		simpl 108	. . . . . . . . . 10 $/\$
81		simpr 109	. . . . . . . . . 10 $/\$
82	80, 81	neeq12d 2329	. . . . . . . . 9 $/\$
83	80	oveq2d 5798	. . . . . . . . . . 11 $/\$
84	83	fveq2d 5433	. . . . . . . . . 10 $/\$
85	81	oveq2d 5798	. . . . . . . . . . 11 $/\$
86	85	fveq2d 5433	. . . . . . . . . 10 $/\$
87	84, 86	breq12d 3950	. . . . . . . . 9 $/\$ # #
88	82, 87	imbi12d 233	. . . . . . . 8 $/\$ # #
89	88	rspc2gv 2805	. . . . . . 7 $/\$ # #
90	76, 79, 89	syl2anc 409	. . . . . 6 $/\$ # $/\$ # #
91	64, 73, 90	mp2d 47	. . . . 5 $/\$ # $/\$ #
92		simpllr 524	. . . . . 6 $/\$ # $/\$ $/\$ #
93		2cnd 8817	. . . . . . . . 9 $/\$ # $/\$ $/\$
94		simplr 520	. . . . . . . . . 10 $/\$ # $/\$ $/\$
95		qcn 9453	. . . . . . . . . 10
96	94, 95	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
97		2ap0 8837	. . . . . . . . . 10 #
98	97	a1i 9	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
99		simpr 109	. . . . . . . . . 10 $/\$ # $/\$ $/\$
100		0z 9089	. . . . . . . . . . . 12
101		zq 9445	. . . . . . . . . . . 12
102	100, 101	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
103		qapne 9458	. . . . . . . . . . 11 $/\$ #
104	94, 102, 103	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$ #
105	99, 104	mpbird 166	. . . . . . . . 9 $/\$ # $/\$ $/\$ #
106	93, 96, 98, 105	mulap0d 8443	. . . . . . . 8 $/\$ # $/\$ $/\$ #
107	14, 15	mp1i 10	. . . . . . . . . . 11 $/\$ # $/\$ $/\$
108		qmulcl 9456	. . . . . . . . . . 11 $/\$
109	107, 94, 108	syl2anc 409	. . . . . . . . . 10 $/\$ # $/\$ $/\$
110		qcn 9453	. . . . . . . . . 10
111	109, 110	syl 14	. . . . . . . . 9 $/\$ # $/\$ $/\$
112		0cnd 7783	. . . . . . . . 9 $/\$ # $/\$ $/\$
113		apsym 8392	. . . . . . . . 9 $/\$ # #
114	111, 112, 113	syl2anc 409	. . . . . . . 8 $/\$ # $/\$ $/\$ # #
115	106, 114	mpbid 146	. . . . . . 7 $/\$ # $/\$ $/\$ #
116		qapne 9458	. . . . . . . 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 2329	. . . . . . . . 9 $/\$
122	119	oveq2d 5798	. . . . . . . . . . 11 $/\$
123	122	fveq2d 5433	. . . . . . . . . 10 $/\$
124	120	oveq2d 5798	. . . . . . . . . . 11 $/\$
125	124	fveq2d 5433	. . . . . . . . . 10 $/\$
126	123, 125	breq12d 3950	. . . . . . . . 9 $/\$ # #
127	121, 126	imbi12d 233	. . . . . . . 8 $/\$ # #
128	127	rspc2gv 2805	. . . . . . 7 $/\$ # #
129	102, 109, 128	syl2anc 409	. . . . . 6 $/\$ # $/\$ $/\$ # #
130	92, 118, 129	mp2d 47	. . . . 5 $/\$ # $/\$ $/\$ #
131	62, 63, 91, 130	apdifflemr 13415	. . . 4 $/\$ # $/\$ #
132	131	ralrimiva 2508	. . 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 1332

wcel 1481

wne 2309

wral 2417 class class class wbr 3937

cfv 5131 (class class class)co 5782

cc 7642

cr 7643

cc0 7644

c1 7645

caddc 7647

cmul 7649

clt 7824

cmin 7957

cneg 7958 # cap 8367

cdiv 8456

c2 8795

cz 9078

cq 9438

cabs 10801

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803

This theorem is referenced by: (None)

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