Theorem divfnzn 10037

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 9660	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
2	1	ad2antrr 492	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> x e. CC )
3		nncn 9321	. . . . . . 7 |- ( y e. NN -> y e. CC )
4	3	ad2antlr 493	. . . . . 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 9342	. . . . . . 7 |- ( y e. NN -> y # 0 )
7	6	ad2antlr 493	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> y # 0 )
8	2, 4, 5, 7	divmulapd 9150	. . . . 5 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> ( ( x / y ) = z <-> ( y x. z ) = x ) )
9	8	riotabidva 6059	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) = ( iota_ z e. CC ( y x. z ) = x ) )
10		eqcom 2240	. . . . . . 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 6043	. . . . 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 9128	. . . . . . . 8 |- ( ( x e. CC /\ y e. NN ) -> ( x / y ) e. CC )
17		reueq 3025	. . . . . . . 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 6057	. . . . . . 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 2316	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) e. CC )
23	9, 22	eqeltrrd 2316	. . 3 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( y x. z ) = x ) e. CC )
24	23	rgen2 2636	. 2 |- A. x e. ZZ A. y e. NN ( iota_ z e. CC ( y x. z ) = x ) e. CC
25		df-div 9011	. . . . 5 |- / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
26	25	reseq1i 5061	. . . 4 |- ( / |` ( ZZ X. NN ) ) = ( ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) ) |` ( ZZ X. NN ) )
27		zsscn 9663	. . . . 5 |- ZZ C_ CC
28		nncn 9321	. . . . . . 7 |- ( x e. NN -> x e. CC )
29		nnne0 9341	. . . . . . 7 |- ( x e. NN -> x =/= 0 )
30		eldifsn 3841	. . . . . . 7 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
31	28, 29, 30	sylanbrc 421	. . . . . 6 |- ( x e. NN -> x e. ( CC \ { 0 } ) )
32	31	ssriv 3252	. . . . 5 |- NN C_ ( CC \ { 0 } )
33		resmpo 6189	. . . . 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 430	. . . 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 2259	. . 3 |- ( / |` ( ZZ X. NN ) ) = ( x e. ZZ , y e. NN |-> ( iota_ z e. CC ( y x. z ) = x ) )
36	35	fnmpo 6442	. 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 )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   E!wreu 2530   \ cdif 3217   C_ wss 3220   {csn 3709   class class class wbr 4131   X. cxp 4774   |` cres 4778   Fn wfn 5375   iota_crio 6040  (class class class)co 6088   e. cmpo 6090   CCcc 8183   0cc0 8185   x. cmul 8190   # cap 8917   / cdiv 9010   NNcn 9313   ZZcz 9655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-z 9656

This theorem is used by:  elq  10038  qnnen  13380