Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldhmsubc | Structured version Visualization version GIF version |
Description: According to df-subc 17524, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17555 and subcss2 17558). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) |
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)) |
Ref | Expression |
---|---|
fldhmsubc | ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3903 | . . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing)) | |
2 | 1 | simprbi 497 | . . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing) |
3 | crngring 19795 | . . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring) |
5 | df-field 19994 | . . . . 5 ⊢ Field = (DivRing ∩ CRing) | |
6 | 4, 5 | eleq2s 2857 | . . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring) |
7 | 6 | rgen 3074 | . . 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 45634 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCat‘𝑈))) |
11 | inss1 4162 | . . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing | |
12 | 5, 11 | eqsstri 3955 | . . . . . 6 ⊢ Field ⊆ DivRing |
13 | sslin 4168 | . . . . . 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 3951 | . . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)) |
18 | 15, 17 | sylibr 233 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶) |
19 | ssidd 3944 | . . . . 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 7284 | . . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦)) | |
22 | 21 | adantl 482 | . . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦)) |
23 | simprl 768 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
24 | simpr 485 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷) | |
25 | 24 | adantl 482 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷) |
26 | ovexd 7310 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V) | |
27 | 20, 22, 23, 25, 26 | ovmpod 7425 | . . . . 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 230 | . . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶 |
31 | 30 | sseli 3917 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶) |
32 | 31 | ad2antrl 725 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶) |
33 | 30 | sseli 3917 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐶) |
34 | 33 | adantl 482 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐶) |
35 | 34 | adantl 482 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐶) |
36 | 29, 22, 32, 35, 26 | ovmpod 7425 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦)) |
37 | 19, 27, 36 | 3sstr4d 3968 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
38 | 37 | ralrimivva 3123 | . . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
39 | ovex 7308 | . . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
40 | 9, 39 | fnmpoi 7910 | . . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷) |
41 | 40 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷)) |
42 | 28, 39 | fnmpoi 7910 | . . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶) |
43 | 42 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶)) |
44 | inex1g 5243 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V) | |
45 | 16, 44 | eqeltrid 2843 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
46 | 41, 43, 45 | isssc 17532 | . . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ⊆cat 𝐽 ↔ (𝐷 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)))) |
47 | 18, 38, 46 | mpbir2and 710 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ⊆cat 𝐽) |
48 | 16, 28 | drhmsubc 45638 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) |
49 | eqid 2738 | . . . 4 ⊢ ((RingCat‘𝑈) ↾cat 𝐽) = ((RingCat‘𝑈) ↾cat 𝐽) | |
50 | 49 | subsubc 17568 | . . 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 710 | 1 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 × cxp 5587 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ⊆cat cssc 17519 ↾cat cresc 17520 Subcatcsubc 17521 Ringcrg 19783 CRingccrg 19784 RingHom crh 19956 DivRingcdr 19991 Fieldcfield 19992 RingCatcringc 45561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-hom 16986 df-cco 16987 df-0g 17152 df-cat 17377 df-cid 17378 df-homf 17379 df-ssc 17522 df-resc 17523 df-subc 17524 df-estrc 17839 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-ghm 18832 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-rnghom 19959 df-drng 19993 df-field 19994 df-ringc 45563 |
This theorem is referenced by: (None) |
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