Theorem fldhmsubcALTV 49157

Description: According to df-subc 17893, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17921 and subcss2 17924). 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 3922	. . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing))
2	1	simprbi 503	. . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing)
3		crngring 20373	. . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring)
4	2, 3	syl 18	. . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring)
5		df-field 20882	. . . . 5 ⊢ Field = (DivRing ∩ CRing)
6	4, 5	eleq2s 2883	. . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring)
7	6	rgen 3083	. . 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 49149	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)))
11		inss1 4189	. . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing
12	5, 11	eqsstri 3984	. . . . . 6 ⊢ Field ⊆ DivRing
13		sslin 4195	. . . . . 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 3968	. . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
18	15, 17	sylibr 237	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶)
19		ssidd 3961	. . . . 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 7428	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
22	21	adantl 487	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23		simprl 783	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷)
24		simpr 490	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
25	24	adantl 487	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷)
26		ovexd 7454	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V)
27	20, 22, 23, 25, 26	ovmpod 7571	. . . . 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 234	. . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶
31	30	sseli 3934	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶)
32	31	ad2antrl 741	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶)
33	30	sseli 3934	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐶)
34	33	adantl 487	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐶)
35	34	adantl 487	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐶)
36	29, 22, 32, 35, 26	ovmpod 7571	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
37	19, 27, 36	3sstr4d 3993	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
38	37	ralrimivva 3210	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
39		ovex 7452	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
40	9, 39	fnmpoi 8073	. . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷)
41	40	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷))
42	28, 39	fnmpoi 8073	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
43	42	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
44		inex1g 5290	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V)
45	16, 44	eqeltrid 2869	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
46	41, 43, 45	isssc 17901	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ⊆cat 𝐽 ↔ (𝐷 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))))
47	18, 38, 46	mpbir2and 726	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ⊆cat 𝐽)
48	16, 28	drhmsubcALTV 49153	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
49		eqid 2765	. . . 4 ⊢ ((RingCatALTV‘𝑈) ↾cat 𝐽) = ((RingCatALTV‘𝑈) ↾cat 𝐽)
50	49	subsubc 17934	. . 3 ⊢ (𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)) → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽)))
51	48, 50	syl 18	. 2 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽)))
52	10, 47, 51	mpbir2and 726	1 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5111   × cxp 5661   Fn wfn 6535  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421   ⊆_cat cssc 17888   ↾_cat cresc 17889  Subcatcsubc 17890  Ringcrg 20361  CRingccrg 20362   RingHom crh 20599  DivRingcdr 20879  Fieldcfield 20880  RingCatALTVcringcALTV 49111

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17231  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-ress 17315  df-plusg 17347  df-hom 17358  df-cco 17359  df-0g 17518  df-cat 17748  df-cid 17749  df-homf 17750  df-ssc 17891  df-resc 17892  df-subc 17893  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-mhm 18880  df-grp 19049  df-ghm 19330  df-mgp 20263  df-ur 20310  df-ring 20363  df-cring 20364  df-rhm 20602  df-drng 20881  df-field 20882  df-ringcALTV 49112

This theorem is used by: (None)