Theorem infxrpnf 39674

Description: Adding plus infinity to a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Assertion

Ref	Expression
infxrpnf	|- ( A C_ RR* -> inf ( ( A u. { +oo } ) , RR* , < ) = inf ( A , RR* , < ) )

Proof of Theorem infxrpnf

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

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( A C_ RR* -> A C_ RR* )
2		pnfxr 10092	. . . . . 6 |- +oo e. RR*
3		snssi 4339	. . . . . 6 |- ( +oo e. RR* -> { +oo } C_ RR* )
4	2, 3	ax-mp 5	. . . . 5 |- { +oo } C_ RR*
5	4	a1i 11	. . . 4 |- ( A C_ RR* -> { +oo } C_ RR* )
6	1, 5	unssd 3789	. . 3 |- ( A C_ RR* -> ( A u. { +oo } ) C_ RR* )
7	6	infxrcld 39612	. 2 |- ( A C_ RR* -> inf ( ( A u. { +oo } ) , RR* , < ) e. RR* )
8		infxrcl 12163	. 2 |- ( A C_ RR* -> inf ( A , RR* , < ) e. RR* )
9		ssun1 3776	. . . 4 |- A C_ ( A u. { +oo } )
10	9	a1i 11	. . 3 |- ( A C_ RR* -> A C_ ( A u. { +oo } ) )
11		infxrss 12169	. . 3 |- ( ( A C_ ( A u. { +oo } ) /\ ( A u. { +oo } ) C_ RR* ) -> inf ( ( A u. { +oo } ) , RR* , < ) <_ inf ( A , RR* , < ) )
12	10, 6, 11	syl2anc 693	. 2 |- ( A C_ RR* -> inf ( ( A u. { +oo } ) , RR* , < ) <_ inf ( A , RR* , < ) )
13		infeq1 8382	. . . . . 6 |- ( A = (/) -> inf ( A , RR* , < ) = inf ( (/) , RR* , < ) )
14		xrinf0 12168	. . . . . . . 8 |- inf ( (/) , RR* , < ) = +oo
15	14, 2	eqeltri 2697	. . . . . . 7 |- inf ( (/) , RR* , < ) e. RR*
16	15	a1i 11	. . . . . 6 |- ( A = (/) -> inf ( (/) , RR* , < ) e. RR* )
17	13, 16	eqeltrd 2701	. . . . 5 |- ( A = (/) -> inf ( A , RR* , < ) e. RR* )
18		xrltso 11974	. . . . . . . . 9 |- < Or RR*
19		infsn 8410	. . . . . . . . 9 |- ( ( < Or RR* /\ +oo e. RR* ) -> inf ( { +oo } , RR* , < ) = +oo )
20	18, 2, 19	mp2an 708	. . . . . . . 8 |- inf ( { +oo } , RR* , < ) = +oo
21	20	eqcomi 2631	. . . . . . 7 |- +oo = inf ( { +oo } , RR* , < )
22	21	a1i 11	. . . . . 6 |- ( A = (/) -> +oo = inf ( { +oo } , RR* , < ) )
23	13, 14	syl6eq 2672	. . . . . 6 |- ( A = (/) -> inf ( A , RR* , < ) = +oo )
24		uneq1 3760	. . . . . . . 8 |- ( A = (/) -> ( A u. { +oo } ) = ( (/) u. { +oo } ) )
25		0un 39215	. . . . . . . . 9 |- ( (/) u. { +oo } ) = { +oo }
26	25	a1i 11	. . . . . . . 8 |- ( A = (/) -> ( (/) u. { +oo } ) = { +oo } )
27	24, 26	eqtrd 2656	. . . . . . 7 |- ( A = (/) -> ( A u. { +oo } ) = { +oo } )
28	27	infeq1d 8383	. . . . . 6 |- ( A = (/) -> inf ( ( A u. { +oo } ) , RR* , < ) = inf ( { +oo } , RR* , < ) )
29	22, 23, 28	3eqtr4d 2666	. . . . 5 |- ( A = (/) -> inf ( A , RR* , < ) = inf ( ( A u. { +oo } ) , RR* , < ) )
30	17, 29	xreqled 39546	. . . 4 |- ( A = (/) -> inf ( A , RR* , < ) <_ inf ( ( A u. { +oo } ) , RR* , < ) )
31	30	adantl 482	. . 3 |- ( ( A C_ RR* /\ A = (/) ) -> inf ( A , RR* , < ) <_ inf ( ( A u. { +oo } ) , RR* , < ) )
32		neqne 2802	. . . 4 |- ( -. A = (/) -> A =/= (/) )
33		nfv 1843	. . . . 5 |- F/ x ( A C_ RR* /\ A =/= (/) )
34		nfv 1843	. . . . 5 |- F/ y ( A C_ RR* /\ A =/= (/) )
35		simpl 473	. . . . 5 |- ( ( A C_ RR* /\ A =/= (/) ) -> A C_ RR* )
36	35, 6	syl 17	. . . . 5 |- ( ( A C_ RR* /\ A =/= (/) ) -> ( A u. { +oo } ) C_ RR* )
37		simpr 477	. . . . . . . 8 |- ( ( A C_ RR* /\ x e. A ) -> x e. A )
38		ssel2 3598	. . . . . . . . 9 |- ( ( A C_ RR* /\ x e. A ) -> x e. RR* )
39		xrleid 11983	. . . . . . . . 9 |- ( x e. RR* -> x <_ x )
40	38, 39	syl 17	. . . . . . . 8 |- ( ( A C_ RR* /\ x e. A ) -> x <_ x )
41		breq1 4656	. . . . . . . . 9 |- ( y = x -> ( y <_ x <-> x <_ x ) )
42	41	rspcev 3309	. . . . . . . 8 |- ( ( x e. A /\ x <_ x ) -> E. y e. A y <_ x )
43	37, 40, 42	syl2anc 693	. . . . . . 7 |- ( ( A C_ RR* /\ x e. A ) -> E. y e. A y <_ x )
44	43	ad4ant14 1293	. . . . . 6 |- ( ( ( ( A C_ RR* /\ A =/= (/) ) /\ x e. ( A u. { +oo } ) ) /\ x e. A ) -> E. y e. A y <_ x )
45		simpll 790	. . . . . . 7 |- ( ( ( ( A C_ RR* /\ A =/= (/) ) /\ x e. ( A u. { +oo } ) ) /\ -. x e. A ) -> ( A C_ RR* /\ A =/= (/) ) )
46		elunnel1 3754	. . . . . . . . 9 |- ( ( x e. ( A u. { +oo } ) /\ -. x e. A ) -> x e. { +oo } )
47		elsni 4194	. . . . . . . . 9 |- ( x e. { +oo } -> x = +oo )
48	46, 47	syl 17	. . . . . . . 8 |- ( ( x e. ( A u. { +oo } ) /\ -. x e. A ) -> x = +oo )
49	48	adantll 750	. . . . . . 7 |- ( ( ( ( A C_ RR* /\ A =/= (/) ) /\ x e. ( A u. { +oo } ) ) /\ -. x e. A ) -> x = +oo )
50		simplr 792	. . . . . . . 8 |- ( ( ( A C_ RR* /\ A =/= (/) ) /\ x = +oo ) -> A =/= (/) )
51		ssel2 3598	. . . . . . . . . . . . 13 |- ( ( A C_ RR* /\ y e. A ) -> y e. RR* )
52		pnfge 11964	. . . . . . . . . . . . 13 |- ( y e. RR* -> y <_ +oo )
53	51, 52	syl 17	. . . . . . . . . . . 12 |- ( ( A C_ RR* /\ y e. A ) -> y <_ +oo )
54	53	adantlr 751	. . . . . . . . . . 11 |- ( ( ( A C_ RR* /\ x = +oo ) /\ y e. A ) -> y <_ +oo )
55		simplr 792	. . . . . . . . . . 11 |- ( ( ( A C_ RR* /\ x = +oo ) /\ y e. A ) -> x = +oo )
56	54, 55	breqtrrd 4681	. . . . . . . . . 10 |- ( ( ( A C_ RR* /\ x = +oo ) /\ y e. A ) -> y <_ x )
57	56	ralrimiva 2966	. . . . . . . . 9 |- ( ( A C_ RR* /\ x = +oo ) -> A. y e. A y <_ x )
58	57	adantlr 751	. . . . . . . 8 |- ( ( ( A C_ RR* /\ A =/= (/) ) /\ x = +oo ) -> A. y e. A y <_ x )
59		r19.2z 4060	. . . . . . . 8 |- ( ( A =/= (/) /\ A. y e. A y <_ x ) -> E. y e. A y <_ x )
60	50, 58, 59	syl2anc 693	. . . . . . 7 |- ( ( ( A C_ RR* /\ A =/= (/) ) /\ x = +oo ) -> E. y e. A y <_ x )
61	45, 49, 60	syl2anc 693	. . . . . 6 |- ( ( ( ( A C_ RR* /\ A =/= (/) ) /\ x e. ( A u. { +oo } ) ) /\ -. x e. A ) -> E. y e. A y <_ x )
62	44, 61	pm2.61dan 832	. . . . 5 |- ( ( ( A C_ RR* /\ A =/= (/) ) /\ x e. ( A u. { +oo } ) ) -> E. y e. A y <_ x )
63	33, 34, 35, 36, 62	infleinf2 39641	. . . 4 |- ( ( A C_ RR* /\ A =/= (/) ) -> inf ( A , RR* , < ) <_ inf ( ( A u. { +oo } ) , RR* , < ) )
64	32, 63	sylan2 491	. . 3 |- ( ( A C_ RR* /\ -. A = (/) ) -> inf ( A , RR* , < ) <_ inf ( ( A u. { +oo } ) , RR* , < ) )
65	31, 64	pm2.61dan 832	. 2 |- ( A C_ RR* -> inf ( A , RR* , < ) <_ inf ( ( A u. { +oo } ) , RR* , < ) )
66	7, 8, 12, 65	xrletrid 11986	1 |- ( A C_ RR* -> inf ( ( A u. { +oo } ) , RR* , < ) = inf ( A , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   Or wor 5034  infcinf 8347   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  infxrpnf2  39693