Theorem qbtwnxr 7460

Description: The rational numbers are dense in RR*: any two extended real numbers have a rational between them.

Assertion

Ref	Expression
qbtwnxr	|- ((A e. RR* /\ B e. RR* /\ A < B) -> E.x e. QQ (A < x /\ x < B))

Distinct variable groups:   x,A   x,B

Proof of Theorem qbtwnxr

Step	Hyp	Ref	Expression
1		qbtwnre 7459	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> E.x e. QQ (A < x /\ x < B))
2	1	3expia 1069	. . . . . 6 |- ((A e. RR /\ B e. RR) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
3		peano2re 6599	. . . . . . . . . . 11 |- (A e. RR -> (A + 1) e. RR)
4		ltp1 6989	. . . . . . . . . . 11 |- (A e. RR -> A < (A + 1))
5		qbtwnre 7459	. . . . . . . . . . 11 |- ((A e. RR /\ (A + 1) e. RR /\ A < (A + 1)) -> E.x e. QQ (A < x /\ x < (A + 1)))
6	3, 4, 5	mpd3an23 1193	. . . . . . . . . 10 |- (A e. RR -> E.x e. QQ (A < x /\ x < (A + 1)))
7		ltpnf 6717	. . . . . . . . . . . . . . 15 |- ((A + 1) e. RR -> (A + 1) < +oo)
8	3, 7	syl 12	. . . . . . . . . . . . . 14 |- (A e. RR -> (A + 1) < +oo)
9	8	adantr 425	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. QQ) -> (A + 1) < +oo)
10		qre 7439	. . . . . . . . . . . . . . . 16 |- (x e. QQ -> x e. RR)
11		rexr 6668	. . . . . . . . . . . . . . . 16 |- (x e. RR -> x e. RR*)
12	10, 11	syl 12	. . . . . . . . . . . . . . 15 |- (x e. QQ -> x e. RR*)
13	12	adantl 424	. . . . . . . . . . . . . 14 |- ((A e. RR /\ x e. QQ) -> x e. RR*)
14		rexr 6668	. . . . . . . . . . . . . . . 16 |- ((A + 1) e. RR -> (A + 1) e. RR*)
15	3, 14	syl 12	. . . . . . . . . . . . . . 15 |- (A e. RR -> (A + 1) e. RR*)
16	15	adantr 425	. . . . . . . . . . . . . 14 |- ((A e. RR /\ x e. QQ) -> (A + 1) e. RR*)
17		pnfxr 6660	. . . . . . . . . . . . . . 15 |- +oo e. RR*
18		xrlttr 6728	. . . . . . . . . . . . . . 15 |- ((x e. RR* /\ (A + 1) e. RR* /\ +oo e. RR*) -> ((x < (A + 1) /\ (A + 1) < +oo) -> x < +oo))
19	17, 18	mp3an3 1180	. . . . . . . . . . . . . 14 |- ((x e. RR* /\ (A + 1) e. RR*) -> ((x < (A + 1) /\ (A + 1) < +oo) -> x < +oo))
20	13, 16, 19	syl11anc 524	. . . . . . . . . . . . 13 |- ((A e. RR /\ x e. QQ) -> ((x < (A + 1) /\ (A + 1) < +oo) -> x < +oo))
21	9, 20	mpan2d 766	. . . . . . . . . . . 12 |- ((A e. RR /\ x e. QQ) -> (x < (A + 1) -> x < +oo))
22	21	anim2d 620	. . . . . . . . . . 11 |- ((A e. RR /\ x e. QQ) -> ((A < x /\ x < (A + 1)) -> (A < x /\ x < +oo)))
23	22	reximdva 2203	. . . . . . . . . 10 |- (A e. RR -> (E.x e. QQ (A < x /\ x < (A + 1)) -> E.x e. QQ (A < x /\ x < +oo)))
24	6, 23	mpd 29	. . . . . . . . 9 |- (A e. RR -> E.x e. QQ (A < x /\ x < +oo))
25	24	adantr 425	. . . . . . . 8 |- ((A e. RR /\ B = +oo) -> E.x e. QQ (A < x /\ x < +oo))
26		breq2 3342	. . . . . . . . . . 11 |- (B = +oo -> (x < B <-> x < +oo))
27	26	anbi2d 678	. . . . . . . . . 10 |- (B = +oo -> ((A < x /\ x < B) <-> (A < x /\ x < +oo)))
28	27	rexbidv 2124	. . . . . . . . 9 |- (B = +oo -> (E.x e. QQ (A < x /\ x < B) <-> E.x e. QQ (A < x /\ x < +oo)))
29	28	adantl 424	. . . . . . . 8 |- ((A e. RR /\ B = +oo) -> (E.x e. QQ (A < x /\ x < B) <-> E.x e. QQ (A < x /\ x < +oo)))
30	25, 29	mpbird 213	. . . . . . 7 |- ((A e. RR /\ B = +oo) -> E.x e. QQ (A < x /\ x < B))
31	30	a1d 15	. . . . . 6 |- ((A e. RR /\ B = +oo) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
32		rexr 6668	. . . . . . . . . 10 |- (A e. RR -> A e. RR*)
33		nltmnf 6722	. . . . . . . . . 10 |- (A e. RR* -> -. A < -oo)
34	32, 33	syl 12	. . . . . . . . 9 |- (A e. RR -> -. A < -oo)
35	34	adantr 425	. . . . . . . 8 |- ((A e. RR /\ B = -oo) -> -. A < -oo)
36		breq2 3342	. . . . . . . . . 10 |- (B = -oo -> (A < B <-> A < -oo))
37	36	notbid 673	. . . . . . . . 9 |- (B = -oo -> (-. A < B <-> -. A < -oo))
38	37	adantl 424	. . . . . . . 8 |- ((A e. RR /\ B = -oo) -> (-. A < B <-> -. A < -oo))
39	35, 38	mpbird 213	. . . . . . 7 |- ((A e. RR /\ B = -oo) -> -. A < B)
40	39	pm2.21d 94	. . . . . 6 |- ((A e. RR /\ B = -oo) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
41	2, 31, 40	3jaodan 1163	. . . . 5 |- ((A e. RR /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
42		elxr 6706	. . . . 5 |- (B e. RR* <-> (B e. RR \/ B = +oo \/ B = -oo))
43	41, 42	sylan2b 501	. . . 4 |- ((A e. RR /\ B e. RR*) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
44		pnfnlt 6721	. . . . . . 7 |- (B e. RR* -> -. +oo < B)
45	44	adantl 424	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> -. +oo < B)
46		breq1 3341	. . . . . . . 8 |- (A = +oo -> (A < B <-> +oo < B))
47	46	notbid 673	. . . . . . 7 |- (A = +oo -> (-. A < B <-> -. +oo < B))
48	47	adantr 425	. . . . . 6 |- ((A = +oo /\ B e. RR*) -> (-. A < B <-> -. +oo < B))
49	45, 48	mpbird 213	. . . . 5 |- ((A = +oo /\ B e. RR*) -> -. A < B)
50	49	pm2.21d 94	. . . 4 |- ((A = +oo /\ B e. RR*) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
51		peano2rem 6605	. . . . . . . . . . 11 |- (B e. RR -> (B - 1) e. RR)
52		id 73	. . . . . . . . . . 11 |- (B e. RR -> B e. RR)
53		ltm1 6993	. . . . . . . . . . 11 |- (B e. RR -> (B - 1) < B)
54		qbtwnre 7459	. . . . . . . . . . 11 |- (((B - 1) e. RR /\ B e. RR /\ (B - 1) < B) -> E.x e. QQ ((B - 1) < x /\ x < B))
55	51, 52, 53, 54	syl111anc 1100	. . . . . . . . . 10 |- (B e. RR -> E.x e. QQ ((B - 1) < x /\ x < B))
56		mnflt 6718	. . . . . . . . . . . . . . 15 |- ((B - 1) e. RR -> -oo < (B - 1))
57	51, 56	syl 12	. . . . . . . . . . . . . 14 |- (B e. RR -> -oo < (B - 1))
58	57	adantr 425	. . . . . . . . . . . . 13 |- ((B e. RR /\ x e. QQ) -> -oo < (B - 1))
59		mnfxr 6662	. . . . . . . . . . . . . . 15 |- -oo e. RR*
60		xrlttr 6728	. . . . . . . . . . . . . . 15 |- (( -oo e. RR* /\ (B - 1) e. RR* /\ x e. RR*) -> (( -oo < (B - 1) /\ (B - 1) < x) -> -oo < x))
61	59, 60	mp3an1 1178	. . . . . . . . . . . . . 14 |- (((B - 1) e. RR* /\ x e. RR*) -> (( -oo < (B - 1) /\ (B - 1) < x) -> -oo < x))
62		rexr 6668	. . . . . . . . . . . . . . 15 |- ((B - 1) e. RR -> (B - 1) e. RR*)
63	51, 62	syl 12	. . . . . . . . . . . . . 14 |- (B e. RR -> (B - 1) e. RR*)
64	61, 63, 12	syl2an 503	. . . . . . . . . . . . 13 |- ((B e. RR /\ x e. QQ) -> (( -oo < (B - 1) /\ (B - 1) < x) -> -oo < x))
65	58, 64	mpand 765	. . . . . . . . . . . 12 |- ((B e. RR /\ x e. QQ) -> ((B - 1) < x -> -oo < x))
66	65	anim1d 619	. . . . . . . . . . 11 |- ((B e. RR /\ x e. QQ) -> (((B - 1) < x /\ x < B) -> ( -oo < x /\ x < B)))
67	66	reximdva 2203	. . . . . . . . . 10 |- (B e. RR -> (E.x e. QQ ((B - 1) < x /\ x < B) -> E.x e. QQ ( -oo < x /\ x < B)))
68	55, 67	mpd 29	. . . . . . . . 9 |- (B e. RR -> E.x e. QQ ( -oo < x /\ x < B))
69	68	adantl 424	. . . . . . . 8 |- ((A = -oo /\ B e. RR) -> E.x e. QQ ( -oo < x /\ x < B))
70		breq1 3341	. . . . . . . . . . 11 |- (A = -oo -> (A < x <-> -oo < x))
71	70	anbi1d 679	. . . . . . . . . 10 |- (A = -oo -> ((A < x /\ x < B) <-> ( -oo < x /\ x < B)))
72	71	rexbidv 2124	. . . . . . . . 9 |- (A = -oo -> (E.x e. QQ (A < x /\ x < B) <-> E.x e. QQ ( -oo < x /\ x < B)))
73	72	adantr 425	. . . . . . . 8 |- ((A = -oo /\ B e. RR) -> (E.x e. QQ (A < x /\ x < B) <-> E.x e. QQ ( -oo < x /\ x < B)))
74	69, 73	mpbird 213	. . . . . . 7 |- ((A = -oo /\ B e. RR) -> E.x e. QQ (A < x /\ x < B))
75	74	a1d 15	. . . . . 6 |- ((A = -oo /\ B e. RR) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
76		1z 7368	. . . . . . . . . 10 |- 1 e. ZZ
77		zq 7440	. . . . . . . . . 10 |- (1 e. ZZ -> 1 e. QQ)
78	76, 77	ax-mp 7	. . . . . . . . 9 |- 1 e. QQ
79		breq2 3342	. . . . . . . . . . 11 |- (x = 1 -> (A < x <-> A < 1))
80		breq1 3341	. . . . . . . . . . 11 |- (x = 1 -> (x < B <-> 1 < B))
81	79, 80	anbi12d 690	. . . . . . . . . 10 |- (x = 1 -> ((A < x /\ x < B) <-> (A < 1 /\ 1 < B)))
82	81	rcla4ev 2381	. . . . . . . . 9 |- ((1 e. QQ /\ (A < 1 /\ 1 < B)) -> E.x e. QQ (A < x /\ x < B))
83	78, 82	mpan 759	. . . . . . . 8 |- ((A < 1 /\ 1 < B) -> E.x e. QQ (A < x /\ x < B))
84		1re 6598	. . . . . . . . . 10 |- 1 e. RR
85		mnflt 6718	. . . . . . . . . 10 |- (1 e. RR -> -oo < 1)
86	84, 85	ax-mp 7	. . . . . . . . 9 |- -oo < 1
87		breq1 3341	. . . . . . . . 9 |- (A = -oo -> (A < 1 <-> -oo < 1))
88	86, 87	mpbiri 211	. . . . . . . 8 |- (A = -oo -> A < 1)
89		ltpnf 6717	. . . . . . . . . 10 |- (1 e. RR -> 1 < +oo)
90	84, 89	ax-mp 7	. . . . . . . . 9 |- 1 < +oo
91		breq2 3342	. . . . . . . . 9 |- (B = +oo -> (1 < B <-> 1 < +oo))
92	90, 91	mpbiri 211	. . . . . . . 8 |- (B = +oo -> 1 < B)
93	83, 88, 92	syl2an 503	. . . . . . 7 |- ((A = -oo /\ B = +oo) -> E.x e. QQ (A < x /\ x < B))
94	93	a1d 15	. . . . . 6 |- ((A = -oo /\ B = +oo) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
95		nltmnf 6722	. . . . . . . . 9 |- ( -oo e. RR* -> -. -oo < -oo)
96	59, 95	ax-mp 7	. . . . . . . 8 |- -. -oo < -oo
97		breq12 3343	. . . . . . . 8 |- ((A = -oo /\ B = -oo) -> (A < B <-> -oo < -oo))
98	96, 97	mtbiri 785	. . . . . . 7 |- ((A = -oo /\ B = -oo) -> -. A < B)
99	98	pm2.21d 94	. . . . . 6 |- ((A = -oo /\ B = -oo) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
100	75, 94, 99	3jaodan 1163	. . . . 5 |- ((A = -oo /\ (B e. RR \/ B = +oo \/ B = -oo)) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
101	100, 42	sylan2b 501	. . . 4 |- ((A = -oo /\ B e. RR*) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
102	43, 50, 101	3jaoian 1162	. . 3 |- (((A e. RR \/ A = +oo \/ A = -oo) /\ B e. RR*) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
103		elxr 6706	. . 3 |- (A e. RR* <-> (A e. RR \/ A = +oo \/ A = -oo))
104	102, 103	sylanb 498	. 2 |- ((A e. RR* /\ B e. RR*) -> (A < B -> E.x e. QQ (A < x /\ x < B)))
105	104	3impia 1064	1 |- ((A e. RR* /\ B e. RR* /\ A < B) -> E.x e. QQ (A < x /\ x < B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106   class class class wbr 3338  (class class class)co 4884  RRcr 6385  1c1 6387   + caddc 6389   - cmin 6445  ZZcz 6451  QQcq 6452   +oocpnf 6650   -oocmnf 6651  RR*cxr 6652   < clt 6653

This theorem is referenced by:  ioo0 7535  qdensere 9027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-n0 7309  df-z 7345  df-q 7436

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