Theorem infrnmptle 39650

Description: An indexed infimum of extended reals is smaller than another indexed infimum of extended reals, when every indexed element is smaller than the corresponding one. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
infrnmptle.x	|- F/ x ph
infrnmptle.b	|- ( ( ph /\ x e. A ) -> B e. RR* )
infrnmptle.c	|- ( ( ph /\ x e. A ) -> C e. RR* )
infrnmptle.l	|- ( ( ph /\ x e. A ) -> B <_ C )

Assertion

Ref	Expression
infrnmptle	|- ( ph -> inf ( ran ( x e. A |-> B ) , RR* , < ) <_ inf ( ran ( x e. A |-> C ) , RR* , < ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   B( x)   C( x)

Proof of Theorem infrnmptle

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

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ y ph
2		nfv 1843	. 2 |- F/ z ph
3		infrnmptle.x	. . 3 |- F/ x ph
4		eqid 2622	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
5		infrnmptle.b	. . 3 |- ( ( ph /\ x e. A ) -> B e. RR* )
6	3, 4, 5	rnmptssd 39385	. 2 |- ( ph -> ran ( x e. A |-> B ) C_ RR* )
7		eqid 2622	. . 3 |- ( x e. A |-> C ) = ( x e. A |-> C )
8		infrnmptle.c	. . 3 |- ( ( ph /\ x e. A ) -> C e. RR* )
9	3, 7, 8	rnmptssd 39385	. 2 |- ( ph -> ran ( x e. A |-> C ) C_ RR* )
10		vex 3203	. . . . . 6 |- y e. _V
11	7	elrnmpt 5372	. . . . . 6 |- ( y e. _V -> ( y e. ran ( x e. A |-> C ) <-> E. x e. A y = C ) )
12	10, 11	ax-mp 5	. . . . 5 |- ( y e. ran ( x e. A |-> C ) <-> E. x e. A y = C )
13	12	biimpi 206	. . . 4 |- ( y e. ran ( x e. A |-> C ) -> E. x e. A y = C )
14	13	adantl 482	. . 3 |- ( ( ph /\ y e. ran ( x e. A |-> C ) ) -> E. x e. A y = C )
15		nfmpt1 4747	. . . . . . 7 |- F/_ x ( x e. A |-> B )
16	15	nfrn 5368	. . . . . 6 |- F/_ x ran ( x e. A |-> B )
17		nfv 1843	. . . . . 6 |- F/ x z <_ y
18	16, 17	nfrex 3007	. . . . 5 |- F/ x E. z e. ran ( x e. A |-> B ) z <_ y
19		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> x e. A )
20	4	elrnmpt1 5374	. . . . . . . . 9 |- ( ( x e. A /\ B e. RR* ) -> B e. ran ( x e. A |-> B ) )
21	19, 5, 20	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> B e. ran ( x e. A |-> B ) )
22	21	3adant3 1081	. . . . . . 7 |- ( ( ph /\ x e. A /\ y = C ) -> B e. ran ( x e. A |-> B ) )
23		infrnmptle.l	. . . . . . . . 9 |- ( ( ph /\ x e. A ) -> B <_ C )
24	23	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ x e. A /\ y = C ) -> B <_ C )
25		id 22	. . . . . . . . . 10 |- ( y = C -> y = C )
26	25	eqcomd 2628	. . . . . . . . 9 |- ( y = C -> C = y )
27	26	3ad2ant3 1084	. . . . . . . 8 |- ( ( ph /\ x e. A /\ y = C ) -> C = y )
28	24, 27	breqtrd 4679	. . . . . . 7 |- ( ( ph /\ x e. A /\ y = C ) -> B <_ y )
29		breq1 4656	. . . . . . . 8 |- ( z = B -> ( z <_ y <-> B <_ y ) )
30	29	rspcev 3309	. . . . . . 7 |- ( ( B e. ran ( x e. A |-> B ) /\ B <_ y ) -> E. z e. ran ( x e. A |-> B ) z <_ y )
31	22, 28, 30	syl2anc 693	. . . . . 6 |- ( ( ph /\ x e. A /\ y = C ) -> E. z e. ran ( x e. A |-> B ) z <_ y )
32	31	3exp 1264	. . . . 5 |- ( ph -> ( x e. A -> ( y = C -> E. z e. ran ( x e. A |-> B ) z <_ y ) ) )
33	3, 18, 32	rexlimd 3026	. . . 4 |- ( ph -> ( E. x e. A y = C -> E. z e. ran ( x e. A |-> B ) z <_ y ) )
34	33	adantr 481	. . 3 |- ( ( ph /\ y e. ran ( x e. A |-> C ) ) -> ( E. x e. A y = C -> E. z e. ran ( x e. A |-> B ) z <_ y ) )
35	14, 34	mpd 15	. 2 |- ( ( ph /\ y e. ran ( x e. A |-> C ) ) -> E. z e. ran ( x e. A |-> B ) z <_ y )
36	1, 2, 6, 9, 35	infleinf2 39641	1 |- ( ph -> inf ( ran ( x e. A |-> B ) , RR* , < ) <_ inf ( ran ( x e. A |-> C ) , RR* , < ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ran crn 5115  infcinf 8347   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:  limsupres  39937