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Theorem ofnegsub 9050
Description: Function analogue of negsub 8335. (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 8065 . . 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 1001 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F : A --> CC )
4 mulcl 8067 . . . 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 8033 . . . . . 6 |- 1 e. CC
76negcli 8355 . . . . 5 |- -u 1 e. CC
87fconst6 5486 . . . 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 1002 . . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G : A --> CC )
11 simp1 1000 . . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> A e. V )
12 inidm 3386 . . 3 |- ( A i^i A ) = A
135, 9, 10, 11, 11, 12off 6183 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( ( A X. { -u 1 } ) oF x. G ) : A --> CC )
14 subcl 8286 . . . 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 6183 . 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) : A --> CC )
17 eqidd 2207 . 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 5435 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
20 eqidd 2207 . . . 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 5727 . . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
2321, 22mulcld 8108 . . . 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 6192 . . 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 8497 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) = -u ( G ` x ) )
2624, 25eqtrd 2239 . 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 5727 . . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
2827, 22negsubd 8404 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F ` x ) - ( G ` x ) ) )
293ffnd 5435 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
3027, 22subcld 8398 . . . 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 6180 . . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF - G ) ` x ) = ( ( F ` x ) - ( G ` x ) ) )
3228, 31eqtr4d 2242 . 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 6184 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 981   = wceq 1373   e. wcel 2177   {csn 3637   X. cxp 4680   -->wf 5275   ` cfv 5279  (class class class)co 5956   oFcof 6168   CCcc 7938   1c1 7941   + caddc 7943   x. cmul 7945   - cmin 8258   -ucneg 8259
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-setind 4592  ax-resscn 8032  ax-1cn 8033  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-cnre 8051
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-of 6170  df-sub 8260  df-neg 8261
This theorem is referenced by:  plysub  15295
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