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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldc | Structured version Visualization version GIF version |
Description: The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (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 |
---|---|
fldc | ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6344 | . . 3 ⊢ (𝑈 ∈ 𝑉 → (RingCat‘𝑈) ∈ V) | |
2 | drhmsubc.j | . . . . 5 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) | |
3 | ovex 6823 | . . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
4 | 2, 3 | fnmpt2i 7389 | . . . 4 ⊢ 𝐽 Fn (𝐶 × 𝐶) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶)) |
6 | fldhmsubc.f | . . . . 5 ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) | |
7 | 6, 3 | fnmpt2i 7389 | . . . 4 ⊢ 𝐹 Fn (𝐷 × 𝐷) |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷)) |
9 | drhmsubc.c | . . . 4 ⊢ 𝐶 = (𝑈 ∩ DivRing) | |
10 | inex1g 4935 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V) | |
11 | 9, 10 | syl5eqel 2854 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
12 | df-field 18960 | . . . . . 6 ⊢ Field = (DivRing ∩ CRing) | |
13 | inss1 3981 | . . . . . 6 ⊢ (DivRing ∩ CRing) ⊆ DivRing | |
14 | 12, 13 | eqsstri 3784 | . . . . 5 ⊢ Field ⊆ DivRing |
15 | sslin 3987 | . . . . 5 ⊢ (Field ⊆ DivRing → (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)) | |
16 | 14, 15 | mp1i 13 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)) |
17 | fldhmsubc.d | . . . 4 ⊢ 𝐷 = (𝑈 ∩ Field) | |
18 | 16, 17, 9 | 3sstr4g 3795 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶) |
19 | 1, 5, 8, 11, 18 | rescabs 16700 | . 2 ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) = ((RingCat‘𝑈) ↾cat 𝐹)) |
20 | 9, 2, 17, 6 | fldcat 42610 | . 2 ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat) |
21 | 19, 20 | eqeltrd 2850 | 1 ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∩ cin 3722 ⊆ wss 3723 × cxp 5247 Fn wfn 6026 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 Catccat 16532 ↾cat cresc 16675 CRingccrg 18756 RingHom crh 18922 DivRingcdr 18957 Fieldcfield 18958 RingCatcringc 42531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-hom 16174 df-cco 16175 df-0g 16310 df-cat 16536 df-cid 16537 df-homf 16538 df-ssc 16677 df-resc 16678 df-subc 16679 df-estrc 16970 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-grp 17633 df-ghm 17866 df-mgp 18698 df-ur 18710 df-ring 18757 df-cring 18758 df-rnghom 18925 df-field 18960 df-ringc 42533 |
This theorem is referenced by: (None) |
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