Theorem psrbagmaxlem1x 15181

Description: Lemma for psrbagmax 15183. All the bags in X are (pointwise) less than S, and thus elements of Z. (Contributed by Jim Kingdon, 27-Oct-2026.)

Hypotheses

Ref	Expression
psrbagmaxlem1.x	|- ( ph -> X C_ ( NN0 ^m I ) )
psrbagmaxlem1.xfi	|- ( ph -> X e. Fin )
psrbagmaxlem1.i	|- ( ph -> I e. Fin )
psrbagmaxlem1a.p	|- ( ph -> P : I --> NN0 )
psrbagmaxlem1a.r	|- ( ph -> R : I --> NN0 )
psrbagmaxlem1a.s	|- S = ( i e. I |-> sup ( { ( P ` i ) , ( R ` i ) } , RR , < ) )
psrbagmaxlem1x.z	|- Z = { q e. ( NN0 ^m I ) | q oR <_ S }
psrbagmaxlem1a.x	|- ( ph -> A. j e. I A. x e. X ( x ` j ) <_ ( P ` j ) )
psrbagmaxlem1a.y	|- ( ph -> A. j e. I A. y e. Y ( y ` j ) <_ ( R ` j ) )
psrbagmaxlem1a.yss	|- ( ph -> Y C_ ( NN0 ^m I ) )
psrbagmaxlem1a.yfi	|- ( ph -> Y e. Fin )

Assertion

Ref	Expression
psrbagmaxlem1x	|- ( ph -> X C_ Z )

Distinct variable groups:   i, I, j   I, q   P, i, j   R, i   S, j   S, q   j, X, x   x, Z   ph, i, j   ph, x   x, q

Allowed substitution hints:   ph( y, q)   P( x, y, q)   R( x, y, j, q)   S( x, y, i)   I( x, y)   X( y, i, q)   Y( x, y, i, j, q)   Z( y, i, j, q)

Proof of Theorem psrbagmaxlem1x

Step	Hyp	Ref	Expression
1		psrbagmaxlem1.x	. . . . 5 |- ( ph -> X C_ ( NN0 ^m I ) )
2	1	sselda 3248	. . . 4 |- ( ( ph /\ x e. X ) -> x e. ( NN0 ^m I ) )
3		elmapi 6949	. . . . . . . . . 10 |- ( x e. ( NN0 ^m I ) -> x : I --> NN0 )
4	2, 3	syl 14	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> x : I --> NN0 )
5	4	ffvelcdmda 5846	. . . . . . . 8 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( x ` j ) e. NN0 )
6	5	nn0red 9630	. . . . . . 7 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( x ` j ) e. RR )
7		psrbagmaxlem1a.p	. . . . . . . . . 10 |- ( ph -> P : I --> NN0 )
8	7	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> P : I --> NN0 )
9		simpr 110	. . . . . . . . 9 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> j e. I )
10	8, 9	ffvelcdmd 5847	. . . . . . . 8 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( P ` j ) e. NN0 )
11	10	nn0red 9630	. . . . . . 7 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( P ` j ) e. RR )
12		psrbagmaxlem1.xfi	. . . . . . . . . . 11 |- ( ph -> X e. Fin )
13		psrbagmaxlem1.i	. . . . . . . . . . 11 |- ( ph -> I e. Fin )
14		psrbagmaxlem1a.r	. . . . . . . . . . 11 |- ( ph -> R : I --> NN0 )
15		psrbagmaxlem1a.s	. . . . . . . . . . 11 |- S = ( i e. I |-> sup ( { ( P ` i ) , ( R ` i ) } , RR , < ) )
16	1, 12, 13, 7, 14, 15	psrbagmaxlem1a 15179	. . . . . . . . . 10 |- ( ph -> S : I --> NN0 )
17	16	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> S : I --> NN0 )
18	17, 9	ffvelcdmd 5847	. . . . . . . 8 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( S ` j ) e. NN0 )
19	18	nn0red 9630	. . . . . . 7 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( S ` j ) e. RR )
20		psrbagmaxlem1a.x	. . . . . . . . . 10 |- ( ph -> A. j e. I A. x e. X ( x ` j ) <_ ( P ` j ) )
21	20	r19.21bi 2638	. . . . . . . . 9 |- ( ( ph /\ j e. I ) -> A. x e. X ( x ` j ) <_ ( P ` j ) )
22	21	r19.21bi 2638	. . . . . . . 8 |- ( ( ( ph /\ j e. I ) /\ x e. X ) -> ( x ` j ) <_ ( P ` j ) )
23	22	an32s 574	. . . . . . 7 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( x ` j ) <_ ( P ` j ) )
24	14	ad2antrr 492	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> R : I --> NN0 )
25	24, 9	ffvelcdmd 5847	. . . . . . . . . 10 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( R ` j ) e. NN0 )
26	25	nn0red 9630	. . . . . . . . 9 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( R ` j ) e. RR )
27		maxle1 11999	. . . . . . . . 9 |- ( ( ( P ` j ) e. RR /\ ( R ` j ) e. RR ) -> ( P ` j ) <_ sup ( { ( P ` j ) , ( R ` j ) } , RR , < ) )
28	11, 26, 27	syl2anc 415	. . . . . . . 8 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( P ` j ) <_ sup ( { ( P ` j ) , ( R ` j ) } , RR , < ) )
29		fveq2 5698	. . . . . . . . . . 11 |- ( i = j -> ( P ` i ) = ( P ` j ) )
30		fveq2 5698	. . . . . . . . . . 11 |- ( i = j -> ( R ` i ) = ( R ` j ) )
31	29, 30	preq12d 3796	. . . . . . . . . 10 |- ( i = j -> { ( P ` i ) , ( R ` i ) } = { ( P ` j ) , ( R ` j ) } )
32	31	supeq1d 7333	. . . . . . . . 9 |- ( i = j -> sup ( { ( P ` i ) , ( R ` i ) } , RR , < ) = sup ( { ( P ` j ) , ( R ` j ) } , RR , < ) )
33		nn0maxcl 12013	. . . . . . . . . 10 |- ( ( ( P ` j ) e. NN0 /\ ( R ` j ) e. NN0 ) -> sup ( { ( P ` j ) , ( R ` j ) } , RR , < ) e. NN0 )
34	10, 25, 33	syl2anc 415	. . . . . . . . 9 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> sup ( { ( P ` j ) , ( R ` j ) } , RR , < ) e. NN0 )
35	15, 32, 9, 34	fvmptd3 5802	. . . . . . . 8 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( S ` j ) = sup ( { ( P ` j ) , ( R ` j ) } , RR , < ) )
36	28, 35	breqtrrd 4159	. . . . . . 7 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( P ` j ) <_ ( S ` j ) )
37	6, 11, 19, 23, 36	letrd 8457	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( x ` j ) <_ ( S ` j ) )
38	37	ralrimiva 2623	. . . . 5 |- ( ( ph /\ x e. X ) -> A. j e. I ( x ` j ) <_ ( S ` j ) )
39		elmapfn 6957	. . . . . . 7 |- ( x e. ( NN0 ^m I ) -> x Fn I )
40	2, 39	syl 14	. . . . . 6 |- ( ( ph /\ x e. X ) -> x Fn I )
41	16	ffnd 5537	. . . . . . 7 |- ( ph -> S Fn I )
42	41	adantr 276	. . . . . 6 |- ( ( ph /\ x e. X ) -> S Fn I )
43	13	adantr 276	. . . . . 6 |- ( ( ph /\ x e. X ) -> I e. Fin )
44		inidm 3440	. . . . . 6 |- ( I i^i I ) = I
45		eqidd 2239	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( x ` j ) = ( x ` j ) )
46		eqidd 2239	. . . . . 6 |- ( ( ( ph /\ x e. X ) /\ j e. I ) -> ( S ` j ) = ( S ` j ) )
47	40, 42, 43, 43, 44, 45, 46	ofrfval 6315	. . . . 5 |- ( ( ph /\ x e. X ) -> ( x oR <_ S <-> A. j e. I ( x ` j ) <_ ( S ` j ) ) )
48	38, 47	mpbird 167	. . . 4 |- ( ( ph /\ x e. X ) -> x oR <_ S )
49		breq1 4134	. . . . 5 |- ( q = x -> ( q oR <_ S <-> x oR <_ S ) )
50		psrbagmaxlem1x.z	. . . . 5 |- Z = { q e. ( NN0 ^m I ) | q oR <_ S }
51	49, 50	elrab2 2985	. . . 4 |- ( x e. Z <-> ( x e. ( NN0 ^m I ) /\ x oR <_ S ) )
52	2, 48, 51	sylanbrc 421	. . 3 |- ( ( ph /\ x e. X ) -> x e. Z )
53	52	ex 115	. 2 |- ( ph -> ( x e. X -> x e. Z ) )
54	53	ssrdv 3254	1 |- ( ph -> X C_ Z )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   C_ wss 3220   {cpr 3710   class class class wbr 4131   |-> cmpt 4193   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   oRcofr 6304   ^m cmap 6927   Fincfn 7028   supcsup 7328   RRcr 8184   < clt 8366   <_ cle 8367   NN0cn0 9574

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-ofr 6306  df-1st 6378  df-2nd 6379  df-recs 6581  df-frec 6667  df-map 6929  df-sup 7330  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9654  df-uz 9937  df-rp 10071  df-seqfrec 10905  df-exp 10996  df-cj 11628  df-re 11629  df-im 11630  df-rsqrt 11785  df-abs 11786

This theorem is used by:  psrbagmax  15183