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Theorem fldhmsubc 46630
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.)
Hypotheses
Ref Expression
drhmsubc.c 𝐶 = (𝑈 ∩ DivRing)
drhmsubc.j 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
fldhmsubc.d 𝐷 = (𝑈 ∩ Field)
fldhmsubc.f 𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))
Assertion
Ref Expression
fldhmsubc (𝑈𝑉𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽)))
Distinct variable groups:   𝐶,𝑟,𝑠   𝑈,𝑟,𝑠   𝑉,𝑟,𝑠   𝐷,𝑟,𝑠
Allowed substitution hints:   𝐹(𝑠,𝑟)   𝐽(𝑠,𝑟)

Proof of Theorem fldhmsubc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3960 . . . . . . 7 (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing))
21simprbi 497 . . . . . 6 (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing)
3 crngring 20026 . . . . . 6 (𝑟 ∈ CRing → 𝑟 ∈ Ring)
42, 3syl 17 . . . . 5 (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring)
5 df-field 20268 . . . . 5 Field = (DivRing ∩ CRing)
64, 5eleq2s 2850 . . . 4 (𝑟 ∈ Field → 𝑟 ∈ Ring)
76rgen 3062 . . 3 𝑟 ∈ Field 𝑟 ∈ Ring
8 fldhmsubc.d . . 3 𝐷 = (𝑈 ∩ Field)
9 fldhmsubc.f . . 3 𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))
107, 8, 9srhmsubc 46622 . 2 (𝑈𝑉𝐹 ∈ (Subcat‘(RingCat‘𝑈)))
11 inss1 4224 . . . . . . 7 (DivRing ∩ CRing) ⊆ DivRing
125, 11eqsstri 4012 . . . . . 6 Field ⊆ DivRing
13 sslin 4230 . . . . . 6 (Field ⊆ DivRing → (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
1412, 13ax-mp 5 . . . . 5 (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing)
1514a1i 11 . . . 4 (𝑈𝑉 → (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
16 drhmsubc.c . . . . 5 𝐶 = (𝑈 ∩ DivRing)
178, 16sseq12i 4008 . . . 4 (𝐷𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
1815, 17sylibr 233 . . 3 (𝑈𝑉𝐷𝐶)
19 ssidd 4001 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦))
209a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠)))
21 oveq12 7402 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
2221adantl 482 . . . . . 6 (((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑟 = 𝑥𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23 simprl 769 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑥𝐷)
24 simpr 485 . . . . . . 7 ((𝑥𝐷𝑦𝐷) → 𝑦𝐷)
2524adantl 482 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑦𝐷)
26 ovexd 7428 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥 RingHom 𝑦) ∈ V)
2720, 22, 23, 25, 26ovmpod 7543 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐹𝑦) = (𝑥 RingHom 𝑦))
28 drhmsubc.j . . . . . . 7 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
2928a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
3014, 17mpbir 230 . . . . . . . 8 𝐷𝐶
3130sseli 3974 . . . . . . 7 (𝑥𝐷𝑥𝐶)
3231ad2antrl 726 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑥𝐶)
3330sseli 3974 . . . . . . . 8 (𝑦𝐷𝑦𝐶)
3433adantl 482 . . . . . . 7 ((𝑥𝐷𝑦𝐷) → 𝑦𝐶)
3534adantl 482 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑦𝐶)
3629, 22, 32, 35, 26ovmpod 7543 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
3719, 27, 363sstr4d 4025 . . . 4 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
3837ralrimivva 3199 . . 3 (𝑈𝑉 → ∀𝑥𝐷𝑦𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
39 ovex 7426 . . . . . 6 (𝑟 RingHom 𝑠) ∈ V
409, 39fnmpoi 8038 . . . . 5 𝐹 Fn (𝐷 × 𝐷)
4140a1i 11 . . . 4 (𝑈𝑉𝐹 Fn (𝐷 × 𝐷))
4228, 39fnmpoi 8038 . . . . 5 𝐽 Fn (𝐶 × 𝐶)
4342a1i 11 . . . 4 (𝑈𝑉𝐽 Fn (𝐶 × 𝐶))
44 inex1g 5312 . . . . 5 (𝑈𝑉 → (𝑈 ∩ DivRing) ∈ V)
4516, 44eqeltrid 2836 . . . 4 (𝑈𝑉𝐶 ∈ V)
4641, 43, 45isssc 17749 . . 3 (𝑈𝑉 → (𝐹cat 𝐽 ↔ (𝐷𝐶 ∧ ∀𝑥𝐷𝑦𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))))
4718, 38, 46mpbir2and 711 . 2 (𝑈𝑉𝐹cat 𝐽)
4816, 28drhmsubc 46626 . . 3 (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
49 eqid 2731 . . . 4 ((RingCat‘𝑈) ↾cat 𝐽) = ((RingCat‘𝑈) ↾cat 𝐽)
5049subsubc 17785 . . 3 (𝐽 ∈ (Subcat‘(RingCat‘𝑈)) → (𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCat‘𝑈)) ∧ 𝐹cat 𝐽)))
5148, 50syl 17 . 2 (𝑈𝑉 → (𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCat‘𝑈)) ∧ 𝐹cat 𝐽)))
5210, 47, 51mpbir2and 711 1 (𝑈𝑉𝐹 ∈ (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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