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Theorem ofnegsub 8981
Description: Function analogue of negsub 8267. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofnegsub |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) )
Proof of Theorem ofnegsub
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 7997 . . 3 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
21adantl 277 . 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
3 simp2 1000 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F : A --> CC )
4 mulcl 7999 . . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
54adantl 277 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
6 ax-1cn 7965 . . . . . 6 |- 1 e. CC
76negcli 8287 . . . . 5 |- -u 1 e. CC
87fconst6 5453 . . . 4 |- ( A X. { -u 1 } ) : A --> CC
98a1i 9 . . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( A X. { -u 1 } ) : A --> CC )
10 simp3 1001 . . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G : A --> CC )
11 simp1 999 . . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> A e. V )
12 inidm 3368 . . 3 |- ( A i^i A ) = A
135, 9, 10, 11, 11, 12off 6143 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( ( A X. { -u 1 } ) oF x. G ) : A --> CC )
14 subcl 8218 . . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x - y ) e. CC )
1514adantl 277 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ ( x e. CC /\ y e. CC ) ) -> ( x - y ) e. CC )
1615, 3, 10, 11, 11, 12off 6143 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) : A --> CC )
17 eqidd 2194 . 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
187a1i 9 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> -u 1 e. CC )
1910ffnd 5404 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
20 eqidd 2194 . . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
217a1i 9 . . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> -u 1 e. CC )
2210ffvelcdmda 5693 . . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
2321, 22mulcld 8040 . . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) e. CC )
2411, 18, 19, 20, 23ofc1g 6151 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( ( A X. { -u 1 } ) oF x. G ) ` x ) = ( -u 1 x. ( G ` x ) ) )
2522mulm1d 8429 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) = -u ( G ` x ) )
2624, 25eqtrd 2226 . 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( ( A X. { -u 1 } ) oF x. G ) ` x ) = -u ( G ` x ) )
273ffvelcdmda 5693 . . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
2827, 22negsubd 8336 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F ` x ) - ( G ` x ) ) )
293ffnd 5404 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
3027, 22subcld 8330 . . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) - ( G ` x ) ) e. CC )
3129, 19, 11, 11, 12, 17, 20, 30ofvalg 6140 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF - G ) ` x ) = ( ( F ` x ) - ( G ` x ) ) )
3228, 31eqtr4d 2229 . 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F oF - G ) ` x ) )
332, 3, 13, 11, 11, 12, 16, 17, 26, 32offeq 6144 1 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   {csn 3618   X. cxp 4657   -->wf 5250   ` cfv 5254  (class class class)co 5918   oFcof 6128   CCcc 7870   1c1 7873   + caddc 7875   x. cmul 7877   - cmin 8190   -ucneg 8191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130  df-sub 8192  df-neg 8193
This theorem is referenced by:  plysub  14899
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