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Mirrors > Home > MPE Home > Th. List > Mathboxes > fldhmsubc | Structured version Visualization version GIF version |
Description: According to df-subc 17741, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17772 and subcss2 17775). 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 3960 | . . . . . . 7 ⊢ (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing)) | |
2 | 1 | simprbi 497 | . . . . . 6 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing) |
3 | crngring 20026 | . . . . . 6 ⊢ (𝑟 ∈ CRing → 𝑟 ∈ Ring) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring) |
5 | df-field 20268 | . . . . 5 ⊢ Field = (DivRing ∩ CRing) | |
6 | 4, 5 | eleq2s 2850 | . . . 4 ⊢ (𝑟 ∈ Field → 𝑟 ∈ Ring) |
7 | 6 | rgen 3062 | . . 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 46622 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘(RingCat‘𝑈))) |
11 | inss1 4224 | . . . . . . 7 ⊢ (DivRing ∩ CRing) ⊆ DivRing | |
12 | 5, 11 | eqsstri 4012 | . . . . . 6 ⊢ Field ⊆ DivRing |
13 | sslin 4230 | . . . . . 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 4008 | . . . 4 ⊢ (𝐷 ⊆ 𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)) |
18 | 15, 17 | sylibr 233 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐷 ⊆ 𝐶) |
19 | ssidd 4001 | . . . . 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 7402 | . . . . . . 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 769 | . . . . . 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 7428 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥 RingHom 𝑦) ∈ V) | |
27 | 20, 22, 23, 25, 26 | ovmpod 7543 | . . . . 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 3974 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ 𝐶) |
32 | 31 | ad2antrl 726 | . . . . . 6 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → 𝑥 ∈ 𝐶) |
33 | 30 | sseli 3974 | . . . . . . . 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 7543 | . . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦)) |
37 | 19, 27, 36 | 3sstr4d 4025 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
38 | 37 | ralrimivva 3199 | . . 3 ⊢ (𝑈 ∈ 𝑉 → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)) |
39 | ovex 7426 | . . . . . 6 ⊢ (𝑟 RingHom 𝑠) ∈ V | |
40 | 9, 39 | fnmpoi 8038 | . . . . 5 ⊢ 𝐹 Fn (𝐷 × 𝐷) |
41 | 40 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐹 Fn (𝐷 × 𝐷)) |
42 | 28, 39 | fnmpoi 8038 | . . . . 5 ⊢ 𝐽 Fn (𝐶 × 𝐶) |
43 | 42 | a1i 11 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐽 Fn (𝐶 × 𝐶)) |
44 | inex1g 5312 | . . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ DivRing) ∈ V) | |
45 | 16, 44 | eqeltrid 2836 | . . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ V) |
46 | 41, 43, 45 | isssc 17749 | . . 3 ⊢ (𝑈 ∈ 𝑉 → (𝐹 ⊆cat 𝐽 ↔ (𝐷 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦)))) |
47 | 18, 38, 46 | mpbir2and 711 | . 2 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ⊆cat 𝐽) |
48 | 16, 28 | drhmsubc 46626 | . . 3 ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) |
49 | eqid 2731 | . . . 4 ⊢ ((RingCat‘𝑈) ↾cat 𝐽) = ((RingCat‘𝑈) ↾cat 𝐽) | |
50 | 49 | subsubc 17785 | . . 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 711 | 1 ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3473 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5141 × cxp 5667 Fn wfn 6527 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 ⊆cat cssc 17736 ↾cat cresc 17737 Subcatcsubc 17738 Ringcrg 20014 CRingccrg 20015 RingHom crh 20198 DivRingcdr 20265 Fieldcfield 20266 RingCatcringc 46549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-hom 17203 df-cco 17204 df-0g 17369 df-cat 17594 df-cid 17595 df-homf 17596 df-ssc 17739 df-resc 17740 df-subc 17741 df-estrc 18056 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mhm 18647 df-grp 18797 df-ghm 19056 df-mgp 19947 df-ur 19964 df-ring 20016 df-cring 20017 df-rnghom 20201 df-drng 20267 df-field 20268 df-ringc 46551 |
This theorem is referenced by: (None) |
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