Theorem divfnzn 9744

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 9379	. . . . . . 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 9046	. . . . . . 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 9067	. . . . . . 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 8887	. . . . 5 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> ( ( x / y ) = z <-> ( y x. z ) = x ) )
9	8	riotabidva 5918	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) = ( iota_ z e. CC ( y x. z ) = x ) )
10		eqcom 2207	. . . . . . 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 5903	. . . . 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 8865	. . . . . . . 8 |- ( ( x e. CC /\ y e. NN ) -> ( x / y ) e. CC )
17		reueq 2972	. . . . . . . 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 5916	. . . . . . 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 2283	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) e. CC )
23	9, 22	eqeltrrd 2283	. . 3 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( y x. z ) = x ) e. CC )
24	23	rgen2 2592	. 2 |- A. x e. ZZ A. y e. NN ( iota_ z e. CC ( y x. z ) = x ) e. CC
25		df-div 8748	. . . . 5 |- / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
26	25	reseq1i 4956	. . . 4 |- ( / |` ( ZZ X. NN ) ) = ( ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) ) |` ( ZZ X. NN ) )
27		zsscn 9382	. . . . 5 |- ZZ C_ CC
28		nncn 9046	. . . . . . 7 |- ( x e. NN -> x e. CC )
29		nnne0 9066	. . . . . . 7 |- ( x e. NN -> x =/= 0 )
30		eldifsn 3760	. . . . . . 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 3197	. . . . 5 |- NN C_ ( CC \ { 0 } )
33		resmpo 6045	. . . . 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 2226	. . 3 |- ( / |` ( ZZ X. NN ) ) = ( x e. ZZ , y e. NN |-> ( iota_ z e. CC ( y x. z ) = x ) )
36	35	fnmpo 6290	. 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 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   E!wreu 2486   \ cdif 3163   C_ wss 3166   {csn 3633   class class class wbr 4045   X. cxp 4674   |` cres 4678   Fn wfn 5267   iota_crio 5900  (class class class)co 5946   e. cmpo 5948   CCcc 7925   0cc0 7927   x. cmul 7932   # cap 8656   / cdiv 8747   NNcn 9038   ZZcz 9374

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-po 4344  df-iso 4345  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-z 9375

This theorem is referenced by:  elq  9745  qnnen  12835