Theorem divfnzn 9635

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 9272	. . . . . . 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 8941	. . . . . . 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 8962	. . . . . . 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 8783	. . . . 5 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> ( ( x / y ) = z <-> ( y x. z ) = x ) )
9	8	riotabidva 5860	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) = ( iota_ z e. CC ( y x. z ) = x ) )
10		eqcom 2189	. . . . . . 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 5846	. . . . 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 8761	. . . . . . . 8 |- ( ( x e. CC /\ y e. NN ) -> ( x / y ) e. CC )
17		reueq 2948	. . . . . . . 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 5858	. . . . . . 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 2265	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) e. CC )
23	9, 22	eqeltrrd 2265	. . 3 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( y x. z ) = x ) e. CC )
24	23	rgen2 2573	. 2 |- A. x e. ZZ A. y e. NN ( iota_ z e. CC ( y x. z ) = x ) e. CC
25		df-div 8644	. . . . 5 |- / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
26	25	reseq1i 4915	. . . 4 |- ( / |` ( ZZ X. NN ) ) = ( ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) ) |` ( ZZ X. NN ) )
27		zsscn 9275	. . . . 5 |- ZZ C_ CC
28		nncn 8941	. . . . . . 7 |- ( x e. NN -> x e. CC )
29		nnne0 8961	. . . . . . 7 |- ( x e. NN -> x =/= 0 )
30		eldifsn 3731	. . . . . . 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 3171	. . . . 5 |- NN C_ ( CC \ { 0 } )
33		resmpo 5986	. . . . 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 2208	. . 3 |- ( / |` ( ZZ X. NN ) ) = ( x e. ZZ , y e. NN |-> ( iota_ z e. CC ( y x. z ) = x ) )
36	35	fnmpo 6217	. 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 1363   e. wcel 2158   =/= wne 2357   A.wral 2465   E!wreu 2467   \ cdif 3138   C_ wss 3141   {csn 3604   class class class wbr 4015   X. cxp 4636   |` cres 4640   Fn wfn 5223   iota_crio 5843  (class class class)co 5888   e. cmpo 5890   CCcc 7823   0cc0 7825   x. cmul 7830   # cap 8552   / cdiv 8643   NNcn 8933   ZZcz 9267

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-z 9268

This theorem is referenced by:  elq  9636  qnnen  12446