Theorem ofrle2add 15180

Description: Adding both sides of two 'less than or equal to' relations. Applying le2add 8780 to functions, pointwise. (Contributed by Jim Kingdon, 7-Nov-2026.)

Hypotheses

Ref	Expression
ofrle2add.a	|- ( ph -> A : I --> RR )
ofrle2add.b	|- ( ph -> B : I --> RR )
ofrle2add.c	|- ( ph -> C : I --> RR )
ofrle2add.d	|- ( ph -> D : I --> RR )
ofrle2add.i	|- ( ph -> I e. V )
ofrle2add.ac	|- ( ph -> A oR <_ C )
ofrle2add.bd	|- ( ph -> B oR <_ D )

Assertion

Ref	Expression
ofrle2add	|- ( ph -> ( A oF + B ) oR <_ ( C oF + D ) )

Proof of Theorem ofrle2add

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofrle2add.a	. . . . 5 |- ( ph -> A : I --> RR )
2	1	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ x e. I ) -> ( A ` x ) e. RR )
3		ofrle2add.b	. . . . 5 |- ( ph -> B : I --> RR )
4	3	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ x e. I ) -> ( B ` x ) e. RR )
5		ofrle2add.c	. . . . 5 |- ( ph -> C : I --> RR )
6	5	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ x e. I ) -> ( C ` x ) e. RR )
7		ofrle2add.d	. . . . 5 |- ( ph -> D : I --> RR )
8	7	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ x e. I ) -> ( D ` x ) e. RR )
9		ofrle2add.ac	. . . . . 6 |- ( ph -> A oR <_ C )
10	1	ffnd 5537	. . . . . . 7 |- ( ph -> A Fn I )
11	5	ffnd 5537	. . . . . . 7 |- ( ph -> C Fn I )
12		ofrle2add.i	. . . . . . 7 |- ( ph -> I e. V )
13		inidm 3440	. . . . . . 7 |- ( I i^i I ) = I
14		eqidd 2239	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( A ` x ) = ( A ` x ) )
15		eqidd 2239	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( C ` x ) = ( C ` x ) )
16	10, 11, 12, 12, 13, 14, 15	ofrfval 6315	. . . . . 6 |- ( ph -> ( A oR <_ C <-> A. x e. I ( A ` x ) <_ ( C ` x ) ) )
17	9, 16	mpbid 147	. . . . 5 |- ( ph -> A. x e. I ( A ` x ) <_ ( C ` x ) )
18	17	r19.21bi 2638	. . . 4 |- ( ( ph /\ x e. I ) -> ( A ` x ) <_ ( C ` x ) )
19		ofrle2add.bd	. . . . . 6 |- ( ph -> B oR <_ D )
20	3	ffnd 5537	. . . . . . 7 |- ( ph -> B Fn I )
21	7	ffnd 5537	. . . . . . 7 |- ( ph -> D Fn I )
22		eqidd 2239	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( B ` x ) = ( B ` x ) )
23		eqidd 2239	. . . . . . 7 |- ( ( ph /\ x e. I ) -> ( D ` x ) = ( D ` x ) )
24	20, 21, 12, 12, 13, 22, 23	ofrfval 6315	. . . . . 6 |- ( ph -> ( B oR <_ D <-> A. x e. I ( B ` x ) <_ ( D ` x ) ) )
25	19, 24	mpbid 147	. . . . 5 |- ( ph -> A. x e. I ( B ` x ) <_ ( D ` x ) )
26	25	r19.21bi 2638	. . . 4 |- ( ( ph /\ x e. I ) -> ( B ` x ) <_ ( D ` x ) )
27	2, 4, 6, 8, 18, 26	le2addd 8900	. . 3 |- ( ( ph /\ x e. I ) -> ( ( A ` x ) + ( B ` x ) ) <_ ( ( C ` x ) + ( D ` x ) ) )
28	27	ralrimiva 2623	. 2 |- ( ph -> A. x e. I ( ( A ` x ) + ( B ` x ) ) <_ ( ( C ` x ) + ( D ` x ) ) )
29	2, 4	readdcld 8362	. . 3 |- ( ( ph /\ x e. I ) -> ( ( A ` x ) + ( B ` x ) ) e. RR )
30	6, 8	readdcld 8362	. . 3 |- ( ( ph /\ x e. I ) -> ( ( C ` x ) + ( D ` x ) ) e. RR )
31	10, 20, 12, 12, 13, 14, 22	offval 6314	. . 3 |- ( ph -> ( A oF + B ) = ( x e. I |-> ( ( A ` x ) + ( B ` x ) ) ) )
32	11, 21, 12, 12, 13, 15, 23	offval 6314	. . 3 |- ( ph -> ( C oF + D ) = ( x e. I |-> ( ( C ` x ) + ( D ` x ) ) ) )
33	12, 29, 30, 31, 32	ofrfval2 6323	. 2 |- ( ph -> ( ( A oF + B ) oR <_ ( C oF + D ) <-> A. x e. I ( ( A ` x ) + ( B ` x ) ) <_ ( ( C ` x ) + ( D ` x ) ) ) )
34	28, 33	mpbird 167	1 |- ( ph -> ( A oF + B ) oR <_ ( C oF + D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528   class class class wbr 4131   -->wf 5376   ` cfv 5380  (class class class)co 6088   oFcof 6303   oRcofr 6304   RRcr 8185   + caddc 8189   <_ cle 8368

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-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-ltwlin 8299  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-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-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373

This theorem is used by:  mplsubrglemcl  15249