Theorem fldhmsubcALTV 41895

Description: According to df-subc 16295, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16323 and subcss2 16326). 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 3758	. . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing))
2	1	simprbi 479	. . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing)
3		crngring 18381	. . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring)
4	2, 3	syl 17	. . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring)
5		df-field 18573	. . . . 5 ⊢ Field = (DivRing ∩ CRing)
6	4, 5	eleq2s 2706	. . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring)
7	6	rgen 2906	. . 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 41887	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)))
11		inss1 3795	. . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing
12	5, 11	eqsstri 3598	. . . . . 6 ⊢ Field ⊆ DivRing
13		sslin 3801	. . . . . 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 3594	. . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
18	15, 17	sylibr 223	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶)
19		ssid 3587	. . . . . 6 ⊢ (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
20	19	a1i 11	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦))
21	9	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)))
22		oveq12 6558	. . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23	22	adantl 481	. . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24		simprl 790	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷)
25		simpr 476	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
26	25	adantl 481	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷)
27		ovex 6577	. . . . . . 7 ⊢ (𝑥 RingHom 𝑦) ∈ V
28	27	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V)
29	21, 23, 24, 26, 28	ovmpt2d 6686	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) = (𝑥 RingHom 𝑦))
30		drhmsubcALTV.j	. . . . . . 7 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))
31	30	a1i 11	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)))
32	14, 17	mpbir 220	. . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶
33	32	sseli 3564	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶)
34	33	ad2antrl 760	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶)
35	32	sseli 3564	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐶)
36	35	adantl 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐶)
37	36	adantl 481	. . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐶)
38	31, 23, 34, 37, 28	ovmpt2d 6686	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
39	20, 29, 38	3sstr4d 3611	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
40	39	ralrimivva 2954	. . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
41		ovex 6577	. . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V
42	9, 41	fnmpt2i 7128	. . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷)
43	42	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷))
44	30, 41	fnmpt2i 7128	. . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶)
45	44	a1i 11	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶))
46		inex1g 4729	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V)
47	16, 46	syl5eqel 2692	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V)
48	43, 45, 47	isssc 16303	. . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ⊆cat 𝐽 ↔ (𝐷 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))))
49	18, 40, 48	mpbir2and 959	. 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ⊆cat 𝐽)
50	16, 30	drhmsubcALTV 41891	. . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
51		eqid 2610	. . . 4 ⊢ ((RingCatALTV‘𝑈) ↾cat 𝐽) = ((RingCatALTV‘𝑈) ↾cat 𝐽)
52	51	subsubc 16336	. . 3 ⊢ (𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)) → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽)))
53	50, 52	syl 17	. 2 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽)))
54	10, 49, 53	mpbir2and 959	1 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ⊆_cat cssc 16290   ↾_cat cresc 16291  Subcatcsubc 16292  Ringcrg 18370  CRingccrg 18371   RingHom crh 18535  DivRingcdr 18570  Fieldcfield 18571  RingCatALTVcringcALTV 41796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-hom 15793  df-cco 15794  df-0g 15925  df-cat 16152  df-cid 16153  df-homf 16154  df-ssc 16293  df-resc 16294  df-subc 16295  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-rnghom 18538  df-drng 18572  df-field 18573  df-ringcALTV 41798

This theorem is referenced by: (None)