fldhmsubc - Metamath Proof Explorer

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Theorem fldhmsubc 20694

Description: According to df-subc 17774, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17802 and subcss2 17805). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)

**Hypotheses**
Ref	Expression
drhmsubc.c	⊢ 𝐶 = (𝑈 ∩ DivRing)
drhmsubc.j	⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))
fldhmsubc.d	⊢ 𝐷 = (𝑈 ∩ Field)
fldhmsubc.f	⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))

**Assertion**
Ref	Expression
fldhmsubc	⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘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 fldhmsubc Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3930	. . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing))
2	1	simprbi 496	. . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing)
3		crngring 20154	. . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring)
4	2, 3	syl 17	. . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring)
5		df-field 20641	. . . . 5 ⊢ Field = (DivRing ∩ CRing)
6	4, 5	eleq2s 2846	. . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring)
7	6	rgen 3046	. . 3 ⊢ ∀𝑟 ∈ Field 𝑟 ∈ Ring
8		fldhmsubc.d	. . 3 ⊢ 𝐷 = (𝑈 ∩ Field)
9		fldhmsubc.f	. . 3 ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))
10	7, 8, 9	srhmsubc 20589	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCat‘𝑈)))
11		inss1 4200	. . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing
12	5, 11	eqsstri 3993	. . . . . 6 ⊢ Field ⊆ DivRing
13		sslin 4206	. . . . . 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		drhmsubc.c	. . . . 5 ⊢ 𝐶 = (𝑈 ∩ DivRing)
17	8, 16	sseq12i 3977	. . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
18	15, 17	sylibr 234	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶)
19		ssidd 3970	. . . . 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 7396	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
22	21	adantl 481	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23		simprl 770	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷)
24		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
25	24	adantl 481	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷)
26		ovexd 7422	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V)
27	20, 22, 23, 25, 26	ovmpod 7541	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) = (𝑥 RingHom 𝑦))
28		drhmsubc.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 231	. . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶
31	30	sseli 3942	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶)
32	31	ad2antrl 728	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶)
33	30	sseli 3942	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐶)
34	33	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐶)
35	34	adantl 481	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐶)
36	29, 22, 32, 35, 26	ovmpod 7541	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
37	19, 27, 36	3sstr4d 4002	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
38	37	ralrimivva 3180	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
39		ovex 7420	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
40	9, 39	fnmpoi 8049	. . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷)
41	40	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷))
42	28, 39	fnmpoi 8049	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
43	42	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
44		inex1g 5274	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V)
45	16, 44	eqeltrid 2832	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
46	41, 43, 45	isssc 17782	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ⊆_cat 𝐽 ↔ (𝐷 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))))
47	18, 38, 46	mpbir2and 713	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ⊆_cat 𝐽)
48	16, 28	drhmsubc 20690	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
49		eqid 2729	. . . 4 ⊢ ((RingCat‘𝑈) ↾_cat 𝐽) = ((RingCat‘𝑈) ↾_cat 𝐽)
50	49	subsubc 17815	. . 3 ⊢ (𝐽 ∈ (Subcat‘(RingCat‘𝑈)) → (𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾_cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCat‘𝑈)) ∧ 𝐹 ⊆_cat 𝐽)))
51	48, 50	syl 17	. 2 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾_cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCat‘𝑈)) ∧ 𝐹 ⊆_cat 𝐽)))
52	10, 47, 51	mpbir2and 713	1 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾_cat 𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 class class class wbr 5107 × cxp 5636 Fn wfn 6506 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 ⊆_cat cssc 17769 ↾_cat cresc 17770 Subcatcsubc 17771 Ringcrg 20142 CRingccrg 20143 RingHom crh 20378 RingCatcringc 20554 DivRingcdr 20638 Fieldcfield 20639

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-hom 17244 df-cco 17245 df-0g 17404 df-cat 17629 df-cid 17630 df-homf 17631 df-ssc 17772 df-resc 17773 df-subc 17774 df-estrc 18084 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-grp 18868 df-ghm 19145 df-mgp 20050 df-ur 20091 df-ring 20144 df-cring 20145 df-rhm 20381 df-ringc 20555 df-drng 20640 df-field 20641

This theorem is referenced by: (None)

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