Theorem fldhmsubcALTV 44380

Description: According to df-subc 17084, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17112 and subcss2 17115). 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 ∩ DivRing)
drhmsubcALTV.j	⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))
fldhmsubcALTV.d	⊢ 𝐷 = (𝑈 ∩ Field)
fldhmsubcALTV.f	⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))

Assertion

Ref	Expression
fldhmsubcALTV	⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (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 4171	. . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing))
2	1	simprbi 499	. . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing)
3		crngring 19310	. . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring)
4	2, 3	syl 17	. . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring)
5		df-field 19507	. . . . 5 ⊢ Field = (DivRing ∩ CRing)
6	4, 5	eleq2s 2933	. . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring)
7	6	rgen 3150	. . 3 ⊢ ∀𝑟 ∈ Field 𝑟 ∈ Ring
8		fldhmsubcALTV.d	. . 3 ⊢ 𝐷 = (𝑈 ∩ Field)
9		fldhmsubcALTV.f	. . 3 ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))
10	7, 8, 9	srhmsubcALTV 44372	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)))
11		inss1 4207	. . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing
12	5, 11	eqsstri 4003	. . . . . 6 ⊢ Field ⊆ DivRing
13		sslin 4213	. . . . . 6 ⊢ (Field ⊆ DivRing → (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
14	12, 13	ax-mp 5	. . . . 5 ⊢ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)
15	14	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
16		drhmsubcALTV.c	. . . . 5 ⊢ 𝐶 = (𝑈 ∩ DivRing)
17	8, 16	sseq12i 3999	. . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
18	15, 17	sylibr 236	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶)
19		ssidd 3992	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦))
20	9	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)))
21		oveq12 7167	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
22	21	adantl 484	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23		simprl 769	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷)
24		simpr 487	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
25	24	adantl 484	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷)
26		ovexd 7193	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V)
27	20, 22, 23, 25, 26	ovmpod 7304	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) = (𝑥 RingHom 𝑦))
28		drhmsubcALTV.j	. . . . . . 7 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))
29	28	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
30	14, 17	mpbir 233	. . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶
31	30	sseli 3965	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶)
32	31	ad2antrl 726	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶)
33	30	sseli 3965	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐶)
34	33	adantl 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐶)
35	34	adantl 484	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐶)
36	29, 22, 32, 35, 26	ovmpod 7304	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
37	19, 27, 36	3sstr4d 4016	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
38	37	ralrimivva 3193	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
39		ovex 7191	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
40	9, 39	fnmpoi 7770	. . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷)
41	40	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷))
42	28, 39	fnmpoi 7770	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
43	42	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
44		inex1g 5225	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V)
45	16, 44	eqeltrid 2919	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
46	41, 43, 45	isssc 17092	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ⊆cat 𝐽 ↔ (𝐷 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))))
47	18, 38, 46	mpbir2and 711	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ⊆cat 𝐽)
48	16, 28	drhmsubcALTV 44376	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
49		eqid 2823	. . . 4 ⊢ ((RingCatALTV‘𝑈) ↾cat 𝐽) = ((RingCatALTV‘𝑈) ↾cat 𝐽)
50	49	subsubc 17125	. . 3 ⊢ (𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)) → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽)))
51	48, 50	syl 17	. 2 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽)))
52	10, 47, 51	mpbir2and 711	1 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938   class class class wbr 5068   × cxp 5555   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160   ⊆_cat cssc 17079   ↾_cat cresc 17080  Subcatcsubc 17081  Ringcrg 19299  CRingccrg 19300   RingHom crh 19466  DivRingcdr 19504  Fieldcfield 19505  RingCatALTVcringcALTV 44282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-hom 16591  df-cco 16592  df-0g 16717  df-cat 16941  df-cid 16942  df-homf 16943  df-ssc 17082  df-resc 17083  df-subc 17084  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-grp 18108  df-ghm 18358  df-mgp 19242  df-ur 19254  df-ring 19301  df-cring 19302  df-rnghom 19469  df-drng 19506  df-field 19507  df-ringcALTV 44284

This theorem is referenced by: (None)