Theorem ofpncan3 15181

Description: Subtraction and addition of equals. Applying pncan3 8542 to functions, pointwise. (Contributed by Jim Kingdon, 7-Nov-2026.)

Hypotheses

Ref	Expression
ofpncan3.a	|- ( ph -> A : I --> CC )
ofpncan3.b	|- ( ph -> B : I --> CC )
ofpncan3.i	|- ( ph -> I e. V )

Assertion

Ref	Expression
ofpncan3	|- ( ph -> ( A oF + ( B oF - A ) ) = B )

Proof of Theorem ofpncan3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofpncan3.a	. . . . 5 |- ( ph -> A : I --> CC )
2	1	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ x e. I ) -> ( A ` x ) e. CC )
3		ofpncan3.b	. . . . 5 |- ( ph -> B : I --> CC )
4	3	ffvelcdmda 5846	. . . 4 |- ( ( ph /\ x e. I ) -> ( B ` x ) e. CC )
5	2, 4	pncan3d 8648	. . 3 |- ( ( ph /\ x e. I ) -> ( ( A ` x ) + ( ( B ` x ) - ( A ` x ) ) ) = ( B ` x ) )
6	5	mpteq2dva 4223	. 2 |- ( ph -> ( x e. I |-> ( ( A ` x ) + ( ( B ` x ) - ( A ` x ) ) ) ) = ( x e. I |-> ( B ` x ) ) )
7		ofpncan3.i	. . 3 |- ( ph -> I e. V )
8	4, 2	subcld 8645	. . 3 |- ( ( ph /\ x e. I ) -> ( ( B ` x ) - ( A ` x ) ) e. CC )
9	1	feqmptd 5759	. . 3 |- ( ph -> A = ( x e. I |-> ( A ` x ) ) )
10	3	ffnd 5537	. . . 4 |- ( ph -> B Fn I )
11	1	ffnd 5537	. . . 4 |- ( ph -> A Fn I )
12		inidm 3440	. . . 4 |- ( I i^i I ) = I
13		eqidd 2239	. . . 4 |- ( ( ph /\ x e. I ) -> ( B ` x ) = ( B ` x ) )
14		eqidd 2239	. . . 4 |- ( ( ph /\ x e. I ) -> ( A ` x ) = ( A ` x ) )
15	10, 11, 7, 7, 12, 13, 14	offval 6314	. . 3 |- ( ph -> ( B oF - A ) = ( x e. I |-> ( ( B ` x ) - ( A ` x ) ) ) )
16	7, 2, 8, 9, 15	offval2 6322	. 2 |- ( ph -> ( A oF + ( B oF - A ) ) = ( x e. I |-> ( ( A ` x ) + ( ( B ` x ) - ( A ` x ) ) ) ) )
17	3	feqmptd 5759	. 2 |- ( ph -> B = ( x e. I |-> ( B ` x ) ) )
18	6, 16, 17	3eqtr4d 2281	1 |- ( ph -> ( A oF + ( B oF - A ) ) = B )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   |-> cmpt 4193   -->wf 5376   ` cfv 5380  (class class class)co 6088   oFcof 6303   CCcc 8184   + caddc 8189   - cmin 8505

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-setind 4686  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-distr 8290  ax-i2m1 8291  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297

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-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-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-sub 8507

This theorem is used by:  mplsubrglemcl  15249