Theorem fldhmsubcALTV 40377

Description: According to df-subc 15765, the subcategories (Subcat ` C ) of a category C are subsets of the homomorphisms of C ( see subcssc 15793 and subcss2 15796). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
drhmsubcALTV.c	|- C = ( U i^i DivRing )
drhmsubcALTV.j	|- J = ( r e. C , s e. C |-> ( r RingHom s ) )
fldhmsubcALTV.d	|- D = ( U i^i Field )
fldhmsubcALTV.f	|- F = ( r e. D , s e. D |-> ( r RingHom s ) )

Assertion

Ref	Expression
fldhmsubcALTV	|- ( U e. V -> F e. (Subcat ` ( (RingCatALTV ` U ) |`cat J ) ) )

Distinct variable groups:   C, r, s   U, r, s   V, r, s   D, r, s

Allowed substitution hints:   F( s, r)   J( s, r)

Proof of Theorem fldhmsubcALTV

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

Step	Hyp	Ref	Expression
1		elin 3628	. . . . . . 7 |- ( r e. ( DivRing i^i CRing ) <-> ( r e. DivRing /\ r e. CRing ) )
2	1	simprbi 470	. . . . . 6 |- ( r e. ( DivRing i^i CRing ) -> r e. CRing )
3		crngring 17839	. . . . . 6 |- ( r e. CRing -> r e. Ring )
4	2, 3	syl 17	. . . . 5 |- ( r e. ( DivRing i^i CRing ) -> r e. Ring )
5		df-field 18026	. . . . 5 |- Field = ( DivRing i^i CRing )
6	4, 5	eleq2s 2557	. . . 4 |- ( r e. Field -> r e. Ring )
7	6	rgen 2758	. . 3 |- A. r e. Field r e. Ring
8		fldhmsubcALTV.d	. . 3 |- D = ( U i^i Field )
9		fldhmsubcALTV.f	. . 3 |- F = ( r e. D , s e. D |-> ( r RingHom s ) )
10	7, 8, 9	srhmsubcALTV 40369	. 2 |- ( U e. V -> F e. (Subcat ` (RingCatALTV ` U ) ) )
11		inss1 3663	. . . . . . 7 |- ( DivRing i^i CRing ) C_ DivRing
12	5, 11	eqsstri 3473	. . . . . 6 |- Field C_ DivRing
13		sslin 3669	. . . . . 6 |- (Field C_ DivRing -> ( U i^i Field ) C_ ( U i^i DivRing ) )
14	12, 13	ax-mp 5	. . . . 5 |- ( U i^i Field ) C_ ( U i^i DivRing )
15	14	a1i 11	. . . 4 |- ( U e. V -> ( U i^i Field ) C_ ( U i^i DivRing ) )
16		drhmsubcALTV.c	. . . . 5 |- C = ( U i^i DivRing )
17	8, 16	sseq12i 3469	. . . 4 |- ( D C_ C <-> ( U i^i Field ) C_ ( U i^i DivRing ) )
18	15, 17	sylibr 217	. . 3 |- ( U e. V -> D C_ C )
19		ssid 3462	. . . . . 6 |- ( x RingHom y ) C_ ( x RingHom y )
20	19	a1i 11	. . . . 5 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> ( x RingHom y ) C_ ( x RingHom y ) )
21	9	a1i 11	. . . . . 6 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> F = ( r e. D , s e. D |-> ( r RingHom s ) ) )
22		oveq12 6323	. . . . . . 7 |- ( ( r = x /\ s = y ) -> ( r RingHom s ) = ( x RingHom y ) )
23	22	adantl 472	. . . . . 6 |- ( ( ( U e. V /\ ( x e. D /\ y e. D ) ) /\ ( r = x /\ s = y ) ) -> ( r RingHom s ) = ( x RingHom y ) )
24		simprl 769	. . . . . 6 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> x e. D )
25		simpr 467	. . . . . . 7 |- ( ( x e. D /\ y e. D ) -> y e. D )
26	25	adantl 472	. . . . . 6 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> y e. D )
27		ovex 6342	. . . . . . 7 |- ( x RingHom y ) e. _V
28	27	a1i 11	. . . . . 6 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> ( x RingHom y ) e. _V )
29	21, 23, 24, 26, 28	ovmpt2d 6450	. . . . 5 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> ( x F y ) = ( x RingHom y ) )
30		drhmsubcALTV.j	. . . . . . 7 |- J = ( r e. C , s e. C |-> ( r RingHom s ) )
31	30	a1i 11	. . . . . 6 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> J = ( r e. C , s e. C |-> ( r RingHom s ) ) )
32	14, 17	mpbir 214	. . . . . . . 8 |- D C_ C
33	32	sseli 3439	. . . . . . 7 |- ( x e. D -> x e. C )
34	33	ad2antrl 739	. . . . . 6 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> x e. C )
35	32	sseli 3439	. . . . . . . 8 |- ( y e. D -> y e. C )
36	35	adantl 472	. . . . . . 7 |- ( ( x e. D /\ y e. D ) -> y e. C )
37	36	adantl 472	. . . . . 6 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> y e. C )
38	31, 23, 34, 37, 28	ovmpt2d 6450	. . . . 5 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> ( x J y ) = ( x RingHom y ) )
39	20, 29, 38	3sstr4d 3486	. . . 4 |- ( ( U e. V /\ ( x e. D /\ y e. D ) ) -> ( x F y ) C_ ( x J y ) )
40	39	ralrimivva 2820	. . 3 |- ( U e. V -> A. x e. D A. y e. D ( x F y ) C_ ( x J y ) )
41		ovex 6342	. . . . . 6 |- ( r RingHom s ) e. _V
42	9, 41	fnmpt2i 6888	. . . . 5 |- F Fn ( D X. D )
43	42	a1i 11	. . . 4 |- ( U e. V -> F Fn ( D X. D ) )
44	30, 41	fnmpt2i 6888	. . . . 5 |- J Fn ( C X. C )
45	44	a1i 11	. . . 4 |- ( U e. V -> J Fn ( C X. C ) )
46		inex1g 4559	. . . . 5 |- ( U e. V -> ( U i^i DivRing ) e. _V )
47	16, 46	syl5eqel 2543	. . . 4 |- ( U e. V -> C e. _V )
48	43, 45, 47	isssc 15773	. . 3 |- ( U e. V -> ( F C_cat J <-> ( D C_ C /\ A. x e. D A. y e. D ( x F y ) C_ ( x J y ) ) ) )
49	18, 40, 48	mpbir2and 938	. 2 |- ( U e. V -> F C_cat J )
50	16, 30	drhmsubcALTV 40373	. . 3 |- ( U e. V -> J e. (Subcat ` (RingCatALTV ` U ) ) )
51		eqid 2461	. . . 4 |- ( (RingCatALTV ` U ) |`cat J ) = ( (RingCatALTV ` U ) |`cat J )
52	51	subsubc 15806	. . 3 |- ( J e. (Subcat ` (RingCatALTV ` U ) ) -> ( F e. (Subcat ` ( (RingCatALTV ` U ) |`cat J ) ) <-> ( F e. (Subcat ` (RingCatALTV ` U ) ) /\ F C_cat J ) ) )
53	50, 52	syl 17	. 2 |- ( U e. V -> ( F e. (Subcat ` ( (RingCatALTV ` U ) |`cat J ) ) <-> ( F e. (Subcat ` (RingCatALTV ` U ) ) /\ F C_cat J ) ) )
54	10, 49, 53	mpbir2and 938	1 |- ( U e. V -> F e. (Subcat ` ( (RingCatALTV ` U ) |`cat J ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   _Vcvv 3056   i^i cin 3414   C_ wss 3415   class class class wbr 4415   X. cxp 4850   Fn wfn 5595   ` cfv 5600  (class class class)co 6314   |-> cmpt2 6316   C_<sub>cat</sub> cssc 15760   |`_cat cresc 15761  Subcatcsubc 15762   Ringcrg 17828   CRingccrg 17829   RingHom crh 17988   DivRingcdr 18023  Fieldcfield 18024  RingCatALTVcringcALTV 40278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-pm 7500  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-fz 11813  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-hom 15262  df-cco 15263  df-0g 15388  df-cat 15622  df-cid 15623  df-homf 15624  df-ssc 15763  df-resc 15764  df-subc 15765  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-mhm 16630  df-grp 16721  df-ghm 16929  df-mgp 17772  df-ur 17784  df-ring 17830  df-cring 17831  df-rnghom 17991  df-drng 18025  df-field 18026  df-ringcALTV 40280

This theorem is referenced by: (None)