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 17180, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17208 and subcss2 17211). 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 3857 | . . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing)) | |
2 | 1 | simprbi 500 | . . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing) |
3 | crngring 19421 | . . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring) |
5 | df-field 19617 | . . . . 5 ⊢ Field = (DivRing ∩ CRing) | |
6 | 4, 5 | eleq2s 2851 | . . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring) |
7 | 6 | rgen 3063 | . . 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 45152 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCat‘𝑈))) |
11 | inss1 4117 | . . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing | |
12 | 5, 11 | eqsstri 3909 | . . . . . 6 ⊢ Field ⊆ DivRing |
13 | sslin 4123 | . . . . . 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 3905 | . . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)) |
18 | 15, 17 | sylibr 237 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶) |
19 | ssidd 3898 | . . . . 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 7173 | . . . . . . 7 ⊢ ((𝑟 = 𝑥 ∧ 𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦)) | |
22 | 21 | adantl 485 | . . . . . 6 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (𝑟 = 𝑥 ∧ 𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦)) |
23 | simprl 771 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐷) | |
24 | simpr 488 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷) | |
25 | 24 | adantl 485 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐷) |
26 | ovexd 7199 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V) | |
27 | 20, 22, 23, 25, 26 | ovmpod 7311 | . . . . 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 234 | . . . . . . . 8 ⊢ 𝐷 ⊆ 𝐶 |
31 | 30 | sseli 3871 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶) |
32 | 31 | ad2antrl 728 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶) |
33 | 30 | sseli 3871 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐶) |
34 | 33 | adantl 485 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐶) |
35 | 34 | adantl 485 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑦 ∈ 𝐶) |
36 | 29, 22, 32, 35, 26 | ovmpod 7311 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦)) |
37 | 19, 27, 36 | 3sstr4d 3922 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
38 | 37 | ralrimivva 3103 | . . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
39 | ovex 7197 | . . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
40 | 9, 39 | fnmpoi 7786 | . . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷) |
41 | 40 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷)) |
42 | 28, 39 | fnmpoi 7786 | . . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶) |
43 | 42 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶)) |
44 | inex1g 5184 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V) | |
45 | 16, 44 | eqeltrid 2837 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
46 | 41, 43, 45 | isssc 17188 | . . 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 45156 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) |
49 | eqid 2738 | . . . 4 ⊢ ((RingCat‘𝑈) ↾cat 𝐽) = ((RingCat‘𝑈) ↾cat 𝐽) | |
50 | 49 | subsubc 17221 | . . 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 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∀wral 3053 Vcvv 3397 ∩ cin 3840 ⊆ wss 3841 class class class wbr 5027 × cxp 5517 Fn wfn 6328 ‘cfv 6333 (class class class)co 7164 ∈ cmpo 7166 ⊆cat cssc 17175 ↾cat cresc 17176 Subcatcsubc 17177 Ringcrg 19409 CRingccrg 19410 RingHom crh 19579 DivRingcdr 19614 Fieldcfield 19615 RingCatcringc 45079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-hom 16685 df-cco 16686 df-0g 16811 df-cat 17035 df-cid 17036 df-homf 17037 df-ssc 17178 df-resc 17179 df-subc 17180 df-estrc 17482 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-grp 18215 df-ghm 18467 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-rnghom 19582 df-drng 19616 df-field 19617 df-ringc 45081 |
This theorem is referenced by: (None) |
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