Theorem ltxrlt 7245

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 7220	. . . . 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 3799	. . . 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 3839	. . . 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 182	. . 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 2142	. . . . . . 7 |- ( x = A -> ( x e. RR <-> A e. RR ) )
6		breq1 3796	. . . . . . 7 |- ( x = A -> ( x <RR y <-> A <RR y ) )
7	5, 6	3anbi13d 1246	. . . . . 6 |- ( x = A -> ( ( x e. RR /\ y e. RR /\ x <RR y ) <-> ( A e. RR /\ y e. RR /\ A <RR y ) ) )
8		eleq1 2142	. . . . . . 7 |- ( y = B -> ( y e. RR <-> B e. RR ) )
9		breq2 3797	. . . . . . 7 |- ( y = B -> ( A <RR y <-> A <RR B ) )
10	8, 9	3anbi23d 1247	. . . . . 6 |- ( y = B -> ( ( A e. RR /\ y e. RR /\ A <RR y ) <-> ( A e. RR /\ B e. RR /\ A <RR B ) ) )
11		eqid 2082	. . . . . 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 4032	. . . . 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 941	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A <RR B )
14	12, 13	syl6bi 161	. . . 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 3839	. . . . 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 4401	. . . . . . . . . . 11 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B <-> ( A e. ( RR u. { -oo } ) /\ B e. { +oo } ) )
17	16	simprbi 269	. . . . . . . . . 10 |- ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> B e. { +oo } )
18		elsni 3424	. . . . . . . . . 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 7228	. . . . . . . . 9 |- ( B e. RR -> B =/= +oo )
22	21	neneqd 2267	. . . . . . . 8 |- ( B e. RR -> -. B = +oo )
23		pm2.24 584	. . . . . . . 8 |- ( B = +oo -> ( -. B = +oo -> A <RR B ) )
24	20, 22, 23	syl6ci 1375	. . . . . . 7 |- ( B e. RR -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
25	24	adantl 271	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( ( RR u. { -oo } ) X. { +oo } ) B -> A <RR B ) )
26		brxp 4401	. . . . . . . . . . 11 |- ( A ( { -oo } X. RR ) B <-> ( A e. { -oo } /\ B e. RR ) )
27	26	simplbi 268	. . . . . . . . . 10 |- ( A ( { -oo } X. RR ) B -> A e. { -oo } )
28		elsni 3424	. . . . . . . . . 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 7229	. . . . . . . . 9 |- ( A e. RR -> A =/= -oo )
32	31	neneqd 2267	. . . . . . . 8 |- ( A e. RR -> -. A = -oo )
33		pm2.24 584	. . . . . . . 8 |- ( A = -oo -> ( -. A = -oo -> A <RR B ) )
34	30, 32, 33	syl6ci 1375	. . . . . . 7 |- ( A e. RR -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
35	34	adantr 270	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A ( { -oo } X. RR ) B -> A <RR B ) )
36	25, 35	jaod 670	. . . . 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	syl5bi 150	. . . 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 670	. . 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	syl5bi 150	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <RR B ) )
40	12	3adant3 959	. . . . . 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 175	. . . . 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 685	. . . 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 132	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <RR B ) -> A < B )
44	43	3expia 1141	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A <RR B -> A < B ) )
45	39, 44	impbid 127	1 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 662   /\ w3a 920   = wceq 1285   e. wcel 1434   u. cun 2972   {csn 3406   class class class wbr 3793   {copab 3846   X. cxp 4369   RRcr 7042   <RR cltrr 7047   +oocpnf 7212   -oocmnf 7213   < clt 7215

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-pnf 7217  df-mnf 7218  df-ltxr 7220

This theorem is referenced by:  axltirr  7246  axltwlin  7247  axlttrn  7248  axltadd  7249  axapti  7250  axmulgt0  7251  0lt1  7303  recexre  7745  recexgt0  7747  remulext1  7766  arch  8352  caucvgrelemcau  10004  caucvgre  10005