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Theorem ofnegsub 9257
Description: Function analogue of negsub 8539. (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 8269 . . 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 8271 . . . 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 8237 . . . . . 6 |- 1 e. CC
76negcli 8559 . . . . 5 |- -u 1 e. CC
87fconst6 5573 . . . 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 3434 . . 3 |- ( A i^i A ) = A
135, 9, 10, 11, 11, 12off 6289 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( ( A X. { -u 1 } ) oF x. G ) : A --> CC )
14 subcl 8490 . . . 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 6289 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) : A --> CC )
17 eqidd 2235 . 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 5515 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
20 eqidd 2235 . . . 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 5818 . . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
2321, 22mulcld 8311 . . . 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 6298 . . 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 8702 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) = -u ( G ` x ) )
2624, 25eqtrd 2267 . 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 5818 . . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
2827, 22negsubd 8608 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F ` x ) - ( G ` x ) ) )
293ffnd 5515 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
3027, 22subcld 8602 . . . 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 6286 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF - G ) ` x ) = ( ( F ` x ) - ( G ` x ) ) )
3228, 31eqtr4d 2270 . 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 6290 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 2205   {csn 3695   X. cxp 4753   -->wf 5354   ` cfv 5358  (class class class)co 6059   oFcof 6274   CCcc 8142   1c1 8145   + caddc 8147   x. cmul 8149   - cmin 8462   -ucneg 8463
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 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-setind 4665  ax-resscn 8236  ax-1cn 8237  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-mulcom 8245  ax-addass 8246  ax-mulass 8247  ax-distr 8248  ax-i2m1 8249  ax-1rid 8251  ax-0id 8252  ax-rnegex 8253  ax-cnre 8255
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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4761  df-rel 4762  df-cnv 4763  df-co 4764  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768  df-iota 5318  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-riota 6012  df-ov 6062  df-oprab 6063  df-mpo 6064  df-of 6276  df-sub 8464  df-neg 8465
This theorem is referenced by:  plysub  15749
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