Theorem divfnzn 9580

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 9217	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
2	1	ad2antrr 485	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> x e. CC )
3		nncn 8886	. . . . . . 7 |- ( y e. NN -> y e. CC )
4	3	ad2antlr 486	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> y e. CC )
5		simpr 109	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> z e. CC )
6		nnap0 8907	. . . . . . 7 |- ( y e. NN -> y # 0 )
7	6	ad2antlr 486	. . . . . 6 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> y # 0 )
8	2, 4, 5, 7	divmulapd 8729	. . . . 5 |- ( ( ( x e. ZZ /\ y e. NN ) /\ z e. CC ) -> ( ( x / y ) = z <-> ( y x. z ) = x ) )
9	8	riotabidva 5825	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) = ( iota_ z e. CC ( y x. z ) = x ) )
10		eqcom 2172	. . . . . . 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 5811	. . . . 5 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC z = ( x / y ) ) = ( iota_ z e. CC ( x / y ) = z ) )
13		simpl 108	. . . . . . . . 9 |- ( ( x e. CC /\ y e. NN ) -> x e. CC )
14	3	adantl 275	. . . . . . . . 9 |- ( ( x e. CC /\ y e. NN ) -> y e. CC )
15	6	adantl 275	. . . . . . . . 9 |- ( ( x e. CC /\ y e. NN ) -> y # 0 )
16	13, 14, 15	divclapd 8707	. . . . . . . 8 |- ( ( x e. CC /\ y e. NN ) -> ( x / y ) e. CC )
17		reueq 2929	. . . . . . . 8 |- ( ( x / y ) e. CC <-> E! z e. CC z = ( x / y ) )
18	16, 17	sylib 121	. . . . . . 7 |- ( ( x e. CC /\ y e. NN ) -> E! z e. CC z = ( x / y ) )
19		riotacl 5823	. . . . . . 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 281	. . . . 5 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC z = ( x / y ) ) e. CC )
22	12, 21	eqeltrrd 2248	. . . 4 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( x / y ) = z ) e. CC )
23	9, 22	eqeltrrd 2248	. . 3 |- ( ( x e. ZZ /\ y e. NN ) -> ( iota_ z e. CC ( y x. z ) = x ) e. CC )
24	23	rgen2 2556	. 2 |- A. x e. ZZ A. y e. NN ( iota_ z e. CC ( y x. z ) = x ) e. CC
25		df-div 8590	. . . . 5 |- / = ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) )
26	25	reseq1i 4887	. . . 4 |- ( / |` ( ZZ X. NN ) ) = ( ( x e. CC , y e. ( CC \ { 0 } ) |-> ( iota_ z e. CC ( y x. z ) = x ) ) |` ( ZZ X. NN ) )
27		zsscn 9220	. . . . 5 |- ZZ C_ CC
28		nncn 8886	. . . . . . 7 |- ( x e. NN -> x e. CC )
29		nnne0 8906	. . . . . . 7 |- ( x e. NN -> x =/= 0 )
30		eldifsn 3710	. . . . . . 7 |- ( x e. ( CC \ { 0 } ) <-> ( x e. CC /\ x =/= 0 ) )
31	28, 29, 30	sylanbrc 415	. . . . . 6 |- ( x e. NN -> x e. ( CC \ { 0 } ) )
32	31	ssriv 3151	. . . . 5 |- NN C_ ( CC \ { 0 } )
33		resmpo 5951	. . . . 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 424	. . . 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 2191	. . 3 |- ( / |` ( ZZ X. NN ) ) = ( x e. ZZ , y e. NN |-> ( iota_ z e. CC ( y x. z ) = x ) )
36	35	fnmpo 6181	. 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 103   <-> wb 104   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   E!wreu 2450   \ cdif 3118   C_ wss 3121   {csn 3583   class class class wbr 3989   X. cxp 4609   |` cres 4613   Fn wfn 5193   iota_crio 5808  (class class class)co 5853   e. cmpo 5855   CCcc 7772   0cc0 7774   x. cmul 7779   # cap 8500   / cdiv 8589   NNcn 8878   ZZcz 9212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-z 9213

This theorem is referenced by:  elq  9581  qnnen  12386