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 6685 | . . 3 ⊢ (𝑈 ∈ 𝑉 → (RingCat‘𝑈) ∈ V) | |
2 | drhmsubc.j | . . . . 5 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) | |
3 | ovex 7189 | . . . . 5 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
4 | 2, 3 | fnmpoi 7768 | . . . 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 | fnmpoi 7768 | . . . 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 5223 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V) | |
11 | 9, 10 | eqeltrid 2917 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
12 | df-field 19505 | . . . . . 6 ⊢ Field = (DivRing ∩ CRing) | |
13 | inss1 4205 | . . . . . 6 ⊢ (DivRing ∩ CRing) ⊆ DivRing | |
14 | 12, 13 | eqsstri 4001 | . . . . 5 ⊢ Field ⊆ DivRing |
15 | sslin 4211 | . . . . 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 4012 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶) |
19 | 1, 5, 8, 11, 18 | rescabs 17103 | . 2 ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) = ((RingCat‘𝑈) ↾cat 𝐹)) |
20 | 9, 2, 17, 6 | fldcat 44373 | . 2 ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat) |
21 | 19, 20 | eqeltrd 2913 | 1 ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 × cxp 5553 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Catccat 16935 ↾cat cresc 17078 CRingccrg 19298 RingHom crh 19464 DivRingcdr 19502 Fieldcfield 19503 RingCatcringc 44294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-hom 16589 df-cco 16590 df-0g 16715 df-cat 16939 df-cid 16940 df-homf 16941 df-ssc 17080 df-resc 17081 df-subc 17082 df-estrc 17373 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-rnghom 19467 df-field 19505 df-ringc 44296 |
This theorem is referenced by: (None) |
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