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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldhmsubcALTV | Structured version Visualization version GIF version |
Description: According to df-subc 16472, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16500 and subcss2 16503). 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.) |
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)) |
Ref | Expression |
---|---|
fldhmsubcALTV | ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3796 | . . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing)) | |
2 | 1 | simprbi 480 | . . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing) |
3 | crngring 18558 | . . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring) |
5 | df-field 18750 | . . . . 5 ⊢ Field = (DivRing ∩ CRing) | |
6 | 4, 5 | eleq2s 2719 | . . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring) |
7 | 6 | rgen 2922 | . . 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 42094 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈))) |
11 | inss1 3833 | . . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing | |
12 | 5, 11 | eqsstri 3635 | . . . . . 6 ⊢ Field ⊆ DivRing |
13 | sslin 3839 | . . . . . 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 3631 | . . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)) |
18 | 15, 17 | sylibr 224 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶) |
19 | ssid 3624 | . . . . . 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 6659 | . . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦)) | |
23 | 22 | adantl 482 | . . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦)) |
24 | simprl 794 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
25 | simpr 477 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷) | |
26 | 25 | adantl 482 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷) |
27 | ovexd 6680 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V) | |
28 | 21, 23, 24, 26, 27 | ovmpt2d 6788 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) = (𝑥 RingHom 𝑦)) |
29 | drhmsubcALTV.j | . . . . . . 7 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) | |
30 | 29 | a1i 11 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))) |
31 | 14, 17 | mpbir 221 | . . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶 |
32 | 31 | sseli 3599 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶) |
33 | 32 | ad2antrl 764 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶) |
34 | 31 | sseli 3599 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐶) |
35 | 34 | adantl 482 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐶) |
36 | 35 | adantl 482 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐶) |
37 | 30, 23, 33, 36, 27 | ovmpt2d 6788 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦)) |
38 | 20, 28, 37 | 3sstr4d 3648 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
39 | 38 | ralrimivva 2971 | . . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
40 | ovex 6678 | . . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
41 | 9, 40 | fnmpt2i 7239 | . . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷) |
42 | 41 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷)) |
43 | 29, 40 | fnmpt2i 7239 | . . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶) |
44 | 43 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶)) |
45 | inex1g 4801 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V) | |
46 | 16, 45 | syl5eqel 2705 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
47 | 42, 44, 46 | isssc 16480 | . . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ⊆cat 𝐽 ↔ (𝐷 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)))) |
48 | 18, 39, 47 | mpbir2and 957 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ⊆cat 𝐽) |
49 | 16, 29 | drhmsubcALTV 42098 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈))) |
50 | eqid 2622 | . . . 4 ⊢ ((RingCatALTV‘𝑈) ↾cat 𝐽) = ((RingCatALTV‘𝑈) ↾cat 𝐽) | |
51 | 50 | subsubc 16513 | . . 3 ⊢ (𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)) → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽))) |
52 | 49, 51 | syl 17 | . 2 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹 ⊆cat 𝐽))) |
53 | 10, 48, 52 | mpbir2and 957 | 1 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 × cxp 5112 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ⊆cat cssc 16467 ↾cat cresc 16468 Subcatcsubc 16469 Ringcrg 18547 CRingccrg 18548 RingHom crh 18712 DivRingcdr 18747 Fieldcfield 18748 RingCatALTVcringcALTV 42004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-hom 15966 df-cco 15967 df-0g 16102 df-cat 16329 df-cid 16330 df-homf 16331 df-ssc 16470 df-resc 16471 df-subc 16472 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-ghm 17658 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-rnghom 18715 df-drng 18749 df-field 18750 df-ringcALTV 42006 |
This theorem is referenced by: (None) |
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