Theorem ltxrlt 8085

Description: The standard less-than <RR and the extended real less-than < are identical when restricted to the non-extended reals RR. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltxrlt	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )

Proof of Theorem ltxrlt

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ltxr 8059	. . . . 5 |- < = ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) )
2	1	breqi 4035	. . . 4 |- ( A < B <-> A ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) ) B )
3		brun 4080	. . . 4 |- ( A ( { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } u. ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) ) B <-> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) )
4	2, 3	bitri 184	. . 3 |- ( A < B <-> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) )
5		eleq1 2256	. . . . . . 7 |- ( x = A -> ( x e. RR <-> A e. RR ) )
6		breq1 4032	. . . . . . 7 |- ( x = A -> ( x <RR y <-> A <RR y ) )
7	5, 6	3anbi13d 1325	. . . . . 6 |- ( x = A -> ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( A e. RR /\ y e. RR /\ A <RR y ) ) )
8		eleq1 2256	. . . . . . 7 |- ( y = B -> ( y e. RR <-> B e. RR ) )
9		breq2 4033	. . . . . . 7 |- ( y = B -> ( A <RR y <-> A <RR B ) )
10	8, 9	3anbi23d 1326	. . . . . 6 |- ( y = B -> ( ( A e. RR /\ y e. RR /\ A <RR y ) <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
11		eqid 2193	. . . . . 6 |- { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } = { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) }
12	7, 10, 11	brabg 4299	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
13		simp3 1001	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A <RR B )
14	12, 13	biimtrdi 163	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B -> A <RR B ) )
15		brun 4080	. . . . 5 |- ( A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B <-> ( A ( ( RR u. { -oo } ) X. { +oo } ) B \/ A ( { -oo } X. RR ) B ) )
16		brxp 4690	. . . . . . . . . . 11 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B <-> ( A e. ( RR u. { -oo } ) /\ B e. { +oo } ) )
17	16	simprbi 275	. . . . . . . . . 10 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B e. { +oo } )
18		elsni 3636	. . . . . . . . . 10 |- ( B e. { +oo } -> B = +oo )
19	17, 18	syl 14	. . . . . . . . 9 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B = +oo )
20	19	a1i 9	. . . . . . . 8 |- ( B e. RR -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B = +oo ) )
21		renepnf 8067	. . . . . . . . 9 |- ( B e. RR -> B =/= +oo )
22	21	neneqd 2385	. . . . . . . 8 |- ( B e. RR -> -. B = +oo )
23		pm2.24 622	. . . . . . . 8 |- ( B = +oo -> ( -. B = +oo -> A <RR B ) )
24	20, 22, 23	syl6ci 1456	. . . . . . 7 |- ( B e. RR -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
25	24	adantl 277	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
26		brxp 4690	. . . . . . . . . . 11 |- ( A ( { -oo } X. RR ) B <-> ( A e. { -oo } /\ B e. RR ) )
27	26	simplbi 274	. . . . . . . . . 10 |- ( A ( { -oo } X. RR ) B -> A e. { -oo } )
28		elsni 3636	. . . . . . . . . 10 |- ( A e. { -oo } -> A = -oo )
29	27, 28	syl 14	. . . . . . . . 9 |- ( A ( { -oo } X. RR ) B -> A = -oo )
30	29	a1i 9	. . . . . . . 8 |- ( A e. RR -> ( A ( { -oo } X. RR ) B -> A = -oo ) )
31		renemnf 8068	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
32	31	neneqd 2385	. . . . . . . 8 |- ( A e. RR -> -. A = -oo )
33		pm2.24 622	. . . . . . . 8 |- ( A = -oo -> ( -. A = -oo -> A <RR B ) )
34	30, 32, 33	syl6ci 1456	. . . . . . 7 |- ( A e. RR -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
35	34	adantr 276	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
36	25, 35	jaod 718	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A ( ( RR u. { -oo } ) X. { +oo } ) B \/ A ( { -oo } X. RR ) B ) -> A <RR B ) )
37	15, 36	biimtrid 152	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B -> A <RR B ) )
38	14, 37	jaod 718	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) -> A <RR B ) )
39	4, 38	biimtrid 152	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <RR B ) )
40	12	3adant3 1019	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
41	40	ibir 177	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B )
42	41	orcd 734	. . . 4 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> ( A { <. x , y >. | ( x e. RR /\ y e. RR /\ x <RR y ) } B \/ A ( ( ( RR u. { -oo } ) X. { +oo } ) u. ( { -oo } X. RR ) ) B ) )
43	42, 4	sylibr 134	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A < B )
44	43	3expia 1207	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A <RR B -> A < B ) )
45	39, 44	impbid 129	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   u. cun 3151   {csn 3618   class class class wbr 4029   {copab 4089   X. cxp 4657   RRcr 7871   <RR cltrr 7876   +oocpnf 8051   -oocmnf 8052   < clt 8054

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-pnf 8056  df-mnf 8057  df-ltxr 8059

This theorem is referenced by:  axltirr  8086  axltwlin  8087  axlttrn  8088  axltadd  8089  axapti  8090  axmulgt0  8091  axsuploc  8092  0lt1  8146  recexre  8597  recexgt0  8599  remulext1  8618  arch  9237  caucvgrelemcau  11124  caucvgre  11125