Theorem divfnzn 9712

Description: Division restricted to ZZ X. NN is a function. Given excluded middle, it would be easy to prove this for CC X. ( CC \ { 0 } ). The key difference is that an element of NN is apart from zero, whereas being an element of CC \ { 0 } implies being not equal to zero. (Contributed by Jim Kingdon, 19-Mar-2020.)

Assertion

Ref	Expression
divfnzn	|- ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN )

Proof of Theorem divfnzn

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zcn 9348	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
2	1	ad2antrr 488	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> x e. CC )
3		nncn 9015	. . . . . . 7 |- ( y e. NN -> y e. CC )
4	3	ad2antlr 489	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> y e. CC )
5		simpr 110	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> z e. CC )
6		nnap0 9036	. . . . . . 7 |- ( y e. NN -> y # 0 )
7	6	ad2antlr 489	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> y # 0 )
8	2, 4, 5, 7	divmulapd 8856	. . . . 5 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> ( ( x / y ) = z <-> ( y x. z ) = x ) )
9	8	riotabidva 5897	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) = ( iota_ z e. CC ( y x. z ) = x ) )
10		eqcom 2198	. . . . . . 7 |- ( z = ( x / y ) <-> ( x / y ) = z )
11	10	a1i 9	. . . . . 6 |- ( ( x e. ZZ /\ y e. NN ) -> ( z = ( x / y ) <-> ( x / y ) = z ) )
12	11	riotabidv 5882	. . . . 5 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC z = ( x / y ) ) = ( iota_ z e. CC ( x / y ) = z ) )
13		simpl 109	. . . . . . . . 9 |- ( ( x e. CC /\ y e. NN ) -> x e. CC )
14	3	adantl 277	. . . . . . . . 9 |- ( ( x e. CC /\ y e. NN ) -> y e. CC )
15	6	adantl 277	. . . . . . . . 9 |- ( ( x e. CC /\ y e. NN ) -> y # 0 )
16	13, 14, 15	divclapd 8834	. . . . . . . 8 |- ( ( x e. CC /\ y e. NN ) -> ( x / y ) e. CC )
17		reueq 2963	. . . . . . . 8 |- ( ( x / y ) e. CC <-> E! z e. CC z = ( x / y ) )
18	16, 17	sylib 122	. . . . . . 7 |- ( ( x e. CC /\ y e. NN ) -> E! z e. CC z = ( x / y ) )
19		riotacl 5895	. . . . . . 7 |- ( E! z e. CC z = ( x / y ) -> ( iota_ z e. CC z = ( x / y ) ) e. CC )
20	18, 19	syl 14	. . . . . 6 |- ( ( x e. CC /\ y e. NN ) -> ( iota_ z e. CC z = ( x / y ) ) e. CC )
21	1, 20	sylan 283	. . . . 5 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC z = ( x / y ) ) e. CC )
22	12, 21	eqeltrrd 2274	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) e. CC )
23	9, 22	eqeltrrd 2274	. . 3 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( y x. z ) = x ) e. CC )
24	23	rgen2 2583	. 2 |- A. x e. ZZ A. y e. NN ( iota_ z e. CC ( y x. z ) = x ) e. CC
25		df-div 8717	. . . . 5 |- / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
26	25	reseq1i 4943	. . . 4 |- ( / |` ( ZZ X. NN ) ) = ( ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) ) |` ( ZZ X. NN ) )
27		zsscn 9351	. . . . 5 |- ZZ C_ CC
28		nncn 9015	. . . . . . 7 |- ( x e. NN -> x e. CC )
29		nnne0 9035	. . . . . . 7 |- ( x e. NN -> x =/= 0 )
30		eldifsn 3750	. . . . . . 7 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
31	28, 29, 30	sylanbrc 417	. . . . . 6 |- ( x e. NN -> x e. ( CC \ { 0 } ) )
32	31	ssriv 3188	. . . . 5 |- NN C_ ( CC \ { 0 } )
33		resmpo 6024	. . . . 5 |- ( ( ZZ C_ CC /\ NN C_ ( CC \ { 0 } ) ) -> ( ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) ) |` ( ZZ X. NN ) ) = ( x e. ZZ , y e. NN |-> ( iota_ z e. CC ( y x. z ) = x ) ) )
34	27, 32, 33	mp2an 426	. . . 4 |- ( ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) ) |` ( ZZ X. NN ) ) = ( x e. ZZ , y e. NN |-> ( iota_ z e. CC ( y x. z ) = x ) )
35	26, 34	eqtri 2217	. . 3 |- ( / |` ( ZZ X. NN ) ) = ( x e. ZZ , y e. NN |-> ( iota_ z e. CC ( y x. z ) = x ) )
36	35	fnmpo 6269	. 2 |- ( A. x e. ZZ A. y e. NN ( iota_ z e. CC ( y x. z ) = x ) e. CC -> ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN ) )
37	24, 36	ax-mp 5	1 |- ( / |` ( ZZ X. NN ) ) Fn ( ZZ X. NN )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   E!wreu 2477   \ cdif 3154   C_ wss 3157   {csn 3623   class class class wbr 4034   X. cxp 4662   |` cres 4666   Fn wfn 5254   iota_crio 5879  (class class class)co 5925   e. cmpo 5927   CCcc 7894   0cc0 7896   x. cmul 7901   # cap 8625   / cdiv 8716   NNcn 9007   ZZcz 9343

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-z 9344

This theorem is referenced by:  elq  9713  qnnen  12673