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Theorem ofnegsub 9235
Description: Function analogue of negsub 8520. (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 8251 . . 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 1025 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F : A --> CC )
4 mulcl 8253 . . . 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 8219 . . . . . 6 |- 1 e. CC
76negcli 8540 . . . . 5 |- -u 1 e. CC
87fconst6 5566 . . . 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 1026 . . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G : A --> CC )
11 simp1 1024 . . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> A e. V )
12 inidm 3429 . . 3 |- ( A i^i A ) = A
135, 9, 10, 11, 11, 12off 6278 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( ( A X. { -u 1 } ) oF x. G ) : A --> CC )
14 subcl 8471 . . . 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 6278 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) : A --> CC )
17 eqidd 2233 . 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 5508 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
20 eqidd 2233 . . . 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 5811 . . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
2321, 22mulcld 8293 . . . 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 6287 . . 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 8682 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) = -u ( G ` x ) )
2624, 25eqtrd 2265 . 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 5811 . . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
2827, 22negsubd 8589 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F ` x ) - ( G ` x ) ) )
293ffnd 5508 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
3027, 22subcld 8583 . . . 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 6275 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF - G ) ` x ) = ( ( F ` x ) - ( G ` x ) ) )
3228, 31eqtr4d 2268 . 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 6279 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 1005   = wceq 1398   e. wcel 2203   {csn 3688   X. cxp 4746   -->wf 5347   ` cfv 5351  (class class class)co 6049   oFcof 6263   CCcc 8124   1c1 8127   + caddc 8129   x. cmul 8131   - cmin 8443   -ucneg 8444
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-setind 4658  ax-resscn 8218  ax-1cn 8219  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-sub 8445  df-neg 8446
This theorem is referenced by:  plysub  15610
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