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Theorem fldhmsubc 45160
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.)
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 3857 . . . . . . 7 (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing))
21simprbi 500 . . . . . 6 (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing)
3 crngring 19421 . . . . . 6 (𝑟 ∈ CRing → 𝑟 ∈ Ring)
42, 3syl 17 . . . . 5 (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring)
5 df-field 19617 . . . . 5 Field = (DivRing ∩ CRing)
64, 5eleq2s 2851 . . . 4 (𝑟 ∈ Field → 𝑟 ∈ Ring)
76rgen 3063 . . 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 45152 . 2 (𝑈𝑉𝐹 ∈ (Subcat‘(RingCat‘𝑈)))
11 inss1 4117 . . . . . . 7 (DivRing ∩ CRing) ⊆ DivRing
125, 11eqsstri 3909 . . . . . 6 Field ⊆ DivRing
13 sslin 4123 . . . . . 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 3905 . . . 4 (𝐷𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
1815, 17sylibr 237 . . 3 (𝑈𝑉𝐷𝐶)
19 ssidd 3898 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦))
209a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠)))
21 oveq12 7173 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
2221adantl 485 . . . . . 6 (((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑟 = 𝑥𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
23 simprl 771 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑥𝐷)
24 simpr 488 . . . . . . 7 ((𝑥𝐷𝑦𝐷) → 𝑦𝐷)
2524adantl 485 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑦𝐷)
26 ovexd 7199 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥 RingHom 𝑦) ∈ V)
2720, 22, 23, 25, 26ovmpod 7311 . . . . 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 234 . . . . . . . 8 𝐷𝐶
3130sseli 3871 . . . . . . 7 (𝑥𝐷𝑥𝐶)
3231ad2antrl 728 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑥𝐶)
3330sseli 3871 . . . . . . . 8 (𝑦𝐷𝑦𝐶)
3433adantl 485 . . . . . . 7 ((𝑥𝐷𝑦𝐷) → 𝑦𝐶)
3534adantl 485 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑦𝐶)
3629, 22, 32, 35, 26ovmpod 7311 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
3719, 27, 363sstr4d 3922 . . . 4 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
3837ralrimivva 3103 . . 3 (𝑈𝑉 → ∀𝑥𝐷𝑦𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
39 ovex 7197 . . . . . 6 (𝑟 RingHom 𝑠) ∈ V
409, 39fnmpoi 7786 . . . . 5 𝐹 Fn (𝐷 × 𝐷)
4140a1i 11 . . . 4 (𝑈𝑉𝐹 Fn (𝐷 × 𝐷))
4228, 39fnmpoi 7786 . . . . 5 𝐽 Fn (𝐶 × 𝐶)
4342a1i 11 . . . 4 (𝑈𝑉𝐽 Fn (𝐶 × 𝐶))
44 inex1g 5184 . . . . 5 (𝑈𝑉 → (𝑈 ∩ DivRing) ∈ V)
4516, 44eqeltrid 2837 . . . 4 (𝑈𝑉𝐶 ∈ V)
4641, 43, 45isssc 17188 . . 3 (𝑈𝑉 → (𝐹cat 𝐽 ↔ (𝐷𝐶 ∧ ∀𝑥𝐷𝑦𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))))
4718, 38, 46mpbir2and 713 . 2 (𝑈𝑉𝐹cat 𝐽)
4816, 28drhmsubc 45156 . . 3 (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
49 eqid 2738 . . . 4 ((RingCat‘𝑈) ↾cat 𝐽) = ((RingCat‘𝑈) ↾cat 𝐽)
5049subsubc 17221 . . 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 713 1 (𝑈𝑉𝐹 ∈ (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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