Theorem psrbagleadd1fi 15179

Description: The analogue of " X <_ F implies X + G <_ F + G " (compare leadd1d 8875) for bags. (Contributed by SN, 2-May-2025.) (Revised by Jim Kingdon, 6-Nov-2026.)

Hypotheses

Ref	Expression
psrbagleadd1fi.s	|- S = { y e. ( NN0 ^m I ) | y oR <_ F }
psrbagleadd1fi.t	|- T = { z e. ( NN0 ^m I ) | z oR <_ ( F oF + G ) }
psrbagleadd1fi.f	|- ( ph -> F e. ( NN0 ^m I ) )
psrbagleadd1fi.g	|- ( ph -> G e. ( NN0 ^m I ) )
psrbagleadd1fi.x	|- ( ph -> X e. S )
psrbagleadd1fi.i	|- ( ph -> I e. Fin )

Assertion

Ref	Expression
psrbagleadd1fi	|- ( ph -> ( X oF + G ) e. T )

Distinct variable groups:   y, F   z, F   z, G   y, I   z, I   y, X   z, X

Allowed substitution hints:   ph( y, z)   S( y, z)   T( y, z)   G( y)

Proof of Theorem psrbagleadd1fi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrbagleadd1fi.x	. . . 4 |- ( ph -> X e. S )
2		elrabi 2979	. . . . 5 |- ( X e. { y e. ( NN0 ^m I ) | y oR <_ F } -> X e. ( NN0 ^m I ) )
3		psrbagleadd1fi.s	. . . . 5 |- S = { y e. ( NN0 ^m I ) | y oR <_ F }
4	2, 3	eleq2s 2333	. . . 4 |- ( X e. S -> X e. ( NN0 ^m I ) )
5	1, 4	syl 14	. . 3 |- ( ph -> X e. ( NN0 ^m I ) )
6		psrbagleadd1fi.g	. . 3 |- ( ph -> G e. ( NN0 ^m I ) )
7		psrbagleadd1fi.i	. . 3 |- ( ph -> I e. Fin )
8		psrbagaddclfi 15178	. . 3 |- ( ( X e. ( NN0 ^m I ) /\ G e. ( NN0 ^m I ) /\ I e. Fin ) -> ( X oF + G ) e. ( NN0 ^m I ) )
9	5, 6, 7, 8	syl3anc 1278	. 2 |- ( ph -> ( X oF + G ) e. ( NN0 ^m I ) )
10		elmapi 6949	. . . . . . . 8 |- ( X e. ( NN0 ^m I ) -> X : I --> NN0 )
11	5, 10	syl 14	. . . . . . 7 |- ( ph -> X : I --> NN0 )
12	11	ffvelcdmda 5846	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( X ` x ) e. NN0 )
13	12	nn0red 9631	. . . . 5 |- ( ( ph /\ x e. I ) -> ( X ` x ) e. RR )
14		psrbagleadd1fi.f	. . . . . . . 8 |- ( ph -> F e. ( NN0 ^m I ) )
15		elmapi 6949	. . . . . . . 8 |- ( F e. ( NN0 ^m I ) -> F : I --> NN0 )
16	14, 15	syl 14	. . . . . . 7 |- ( ph -> F : I --> NN0 )
17	16	ffvelcdmda 5846	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. NN0 )
18	17	nn0red 9631	. . . . 5 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. RR )
19		elmapi 6949	. . . . . . . 8 |- ( G e. ( NN0 ^m I ) -> G : I --> NN0 )
20	6, 19	syl 14	. . . . . . 7 |- ( ph -> G : I --> NN0 )
21	20	ffvelcdmda 5846	. . . . . 6 |- ( ( ph /\ x e. I ) -> ( G ` x ) e. NN0 )
22	21	nn0red 9631	. . . . 5 |- ( ( ph /\ x e. I ) -> ( G ` x ) e. RR )
23		breq1 4134	. . . . . . . . . 10 |- ( y = X -> ( y oR <_ F <-> X oR <_ F ) )
24	23, 3	elrab2 2985	. . . . . . . . 9 |- ( X e. S <-> ( X e. ( NN0 ^m I ) /\ X oR <_ F ) )
25	24	simprbi 275	. . . . . . . 8 |- ( X e. S -> X oR <_ F )
26	1, 25	syl 14	. . . . . . 7 |- ( ph -> X oR <_ F )
27	10	ffnd 5537	. . . . . . . . 9 |- ( X e. ( NN0 ^m I ) -> X Fn I )
28	5, 27	syl 14	. . . . . . . 8 |- ( ph -> X Fn I )
29	16	ffnd 5537	. . . . . . . 8 |- ( ph -> F Fn I )
30		inidm 3440	. . . . . . . 8 |- ( I i^i I ) = I
31		eqidd 2239	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> ( X ` x ) = ( X ` x ) )
32		eqidd 2239	. . . . . . . 8 |- ( ( ph /\ x e. I ) -> ( F ` x ) = ( F ` x ) )
33	28, 29, 7, 7, 30, 31, 32	ofrfval 6315	. . . . . . 7 |- ( ph -> ( X oR <_ F <-> A. x e. I ( X ` x ) <_ ( F ` x ) ) )
34	26, 33	mpbid 147	. . . . . 6 |- ( ph -> A. x e. I ( X ` x ) <_ ( F ` x ) )
35	34	r19.21bi 2638	. . . . 5 |- ( ( ph /\ x e. I ) -> ( X ` x ) <_ ( F ` x ) )
36	13, 18, 22, 35	leadd1dd 8895	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( X ` x ) + ( G ` x ) ) <_ ( ( F ` x ) + ( G ` x ) ) )
37	36	ralrimiva 2623	. . 3 |- ( ph -> A. x e. I ( ( X ` x ) + ( G ` x ) ) <_ ( ( F ` x ) + ( G ` x ) ) )
38		elmapi 6949	. . . . . 6 |- ( ( X oF + G ) e. ( NN0 ^m I ) -> ( X oF + G ) : I --> NN0 )
39	9, 38	syl 14	. . . . 5 |- ( ph -> ( X oF + G ) : I --> NN0 )
40	39	ffnd 5537	. . . 4 |- ( ph -> ( X oF + G ) Fn I )
41		psrbagaddclfi 15178	. . . . . . 7 |- ( ( F e. ( NN0 ^m I ) /\ G e. ( NN0 ^m I ) /\ I e. Fin ) -> ( F oF + G ) e. ( NN0 ^m I ) )
42	14, 6, 7, 41	syl3anc 1278	. . . . . 6 |- ( ph -> ( F oF + G ) e. ( NN0 ^m I ) )
43		elmapi 6949	. . . . . 6 |- ( ( F oF + G ) e. ( NN0 ^m I ) -> ( F oF + G ) : I --> NN0 )
44	42, 43	syl 14	. . . . 5 |- ( ph -> ( F oF + G ) : I --> NN0 )
45	44	ffnd 5537	. . . 4 |- ( ph -> ( F oF + G ) Fn I )
46	20	ffnd 5537	. . . . 5 |- ( ph -> G Fn I )
47		eqidd 2239	. . . . 5 |- ( ( ph /\ x e. I ) -> ( G ` x ) = ( G ` x ) )
48	12, 21	nn0addcld 9634	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( X ` x ) + ( G ` x ) ) e. NN0 )
49	28, 46, 7, 7, 30, 31, 47, 48	ofvalg 6316	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( X oF + G ) ` x ) = ( ( X ` x ) + ( G ` x ) ) )
50	17, 21	nn0addcld 9634	. . . . 5 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) + ( G ` x ) ) e. NN0 )
51	29, 46, 7, 7, 30, 32, 47, 50	ofvalg 6316	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F oF + G ) ` x ) = ( ( F ` x ) + ( G ` x ) ) )
52	40, 45, 7, 7, 30, 49, 51	ofrfval 6315	. . 3 |- ( ph -> ( ( X oF + G ) oR <_ ( F oF + G ) <-> A. x e. I ( ( X ` x ) + ( G ` x ) ) <_ ( ( F ` x ) + ( G ` x ) ) ) )
53	37, 52	mpbird 167	. 2 |- ( ph -> ( X oF + G ) oR <_ ( F oF + G ) )
54		breq1 4134	. . 3 |- ( z = ( X oF + G ) -> ( z oR <_ ( F oF + G ) <-> ( X oF + G ) oR <_ ( F oF + G ) ) )
55		psrbagleadd1fi.t	. . 3 |- T = { z e. ( NN0 ^m I ) | z oR <_ ( F oF + G ) }
56	54, 55	elrab2 2985	. 2 |- ( ( X oF + G ) e. T <-> ( ( X oF + G ) e. ( NN0 ^m I ) /\ ( X oF + G ) oR <_ ( F oF + G ) ) )
57	9, 53, 56	sylanbrc 421	1 |- ( ph -> ( X oF + G ) e. T )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   class class class wbr 4131   Fn wfn 5375   -->wf 5376   ` cfv 5380  (class class class)co 6088   oFcof 6303   oRcofr 6304   ^m cmap 6927   Fincfn 7028   + caddc 8189   <_ cle 8368   NN0cn0 9575

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-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-i2m1 8291  ax-0id 8294  ax-rnegex 8295  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  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-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  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-id 4440  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-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-ofr 6306  df-map 6929  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-inn 9315  df-n0 9576

This theorem is used by: (None)