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Theorem fldhmsubc 46456
Description: According to df-subc 17702, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17733 and subcss2 17736). 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 𝐢 = (π‘ˆ ∩ DivRing)
drhmsubc.j 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))
fldhmsubc.d 𝐷 = (π‘ˆ ∩ Field)
fldhmsubc.f 𝐹 = (π‘Ÿ ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (π‘Ÿ RingHom 𝑠))
Assertion
Ref Expression
fldhmsubc (π‘ˆ ∈ 𝑉 β†’ 𝐹 ∈ (Subcatβ€˜((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽)))
Distinct variable groups:   𝐢,π‘Ÿ,𝑠   π‘ˆ,π‘Ÿ,𝑠   𝑉,π‘Ÿ,𝑠   𝐷,π‘Ÿ,𝑠
Allowed substitution hints:   𝐹(𝑠,π‘Ÿ)   𝐽(𝑠,π‘Ÿ)

Proof of Theorem fldhmsubc
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3931 . . . . . . 7 (π‘Ÿ ∈ (DivRing ∩ CRing) ↔ (π‘Ÿ ∈ DivRing ∧ π‘Ÿ ∈ CRing))
21simprbi 498 . . . . . 6 (π‘Ÿ ∈ (DivRing ∩ CRing) β†’ π‘Ÿ ∈ CRing)
3 crngring 19983 . . . . . 6 (π‘Ÿ ∈ CRing β†’ π‘Ÿ ∈ Ring)
42, 3syl 17 . . . . 5 (π‘Ÿ ∈ (DivRing ∩ CRing) β†’ π‘Ÿ ∈ Ring)
5 df-field 20202 . . . . 5 Field = (DivRing ∩ CRing)
64, 5eleq2s 2856 . . . 4 (π‘Ÿ ∈ Field β†’ π‘Ÿ ∈ Ring)
76rgen 3067 . . 3 βˆ€π‘Ÿ ∈ Field π‘Ÿ ∈ Ring
8 fldhmsubc.d . . 3 𝐷 = (π‘ˆ ∩ Field)
9 fldhmsubc.f . . 3 𝐹 = (π‘Ÿ ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (π‘Ÿ RingHom 𝑠))
107, 8, 9srhmsubc 46448 . 2 (π‘ˆ ∈ 𝑉 β†’ 𝐹 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)))
11 inss1 4193 . . . . . . 7 (DivRing ∩ CRing) βŠ† DivRing
125, 11eqsstri 3983 . . . . . 6 Field βŠ† DivRing
13 sslin 4199 . . . . . 6 (Field βŠ† DivRing β†’ (π‘ˆ ∩ Field) βŠ† (π‘ˆ ∩ DivRing))
1412, 13ax-mp 5 . . . . 5 (π‘ˆ ∩ Field) βŠ† (π‘ˆ ∩ DivRing)
1514a1i 11 . . . 4 (π‘ˆ ∈ 𝑉 β†’ (π‘ˆ ∩ Field) βŠ† (π‘ˆ ∩ DivRing))
16 drhmsubc.c . . . . 5 𝐢 = (π‘ˆ ∩ DivRing)
178, 16sseq12i 3979 . . . 4 (𝐷 βŠ† 𝐢 ↔ (π‘ˆ ∩ Field) βŠ† (π‘ˆ ∩ DivRing))
1815, 17sylibr 233 . . 3 (π‘ˆ ∈ 𝑉 β†’ 𝐷 βŠ† 𝐢)
19 ssidd 3972 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯ RingHom 𝑦) βŠ† (π‘₯ RingHom 𝑦))
209a1i 11 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ 𝐹 = (π‘Ÿ ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (π‘Ÿ RingHom 𝑠)))
21 oveq12 7371 . . . . . . 7 ((π‘Ÿ = π‘₯ ∧ 𝑠 = 𝑦) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom 𝑦))
2221adantl 483 . . . . . 6 (((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) ∧ (π‘Ÿ = π‘₯ ∧ 𝑠 = 𝑦)) β†’ (π‘Ÿ RingHom 𝑠) = (π‘₯ RingHom 𝑦))
23 simprl 770 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ π‘₯ ∈ 𝐷)
24 simpr 486 . . . . . . 7 ((π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) β†’ 𝑦 ∈ 𝐷)
2524adantl 483 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ 𝑦 ∈ 𝐷)
26 ovexd 7397 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯ RingHom 𝑦) ∈ V)
2720, 22, 23, 25, 26ovmpod 7512 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯𝐹𝑦) = (π‘₯ RingHom 𝑦))
28 drhmsubc.j . . . . . . 7 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))
2928a1i 11 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ 𝐽 = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠)))
3014, 17mpbir 230 . . . . . . . 8 𝐷 βŠ† 𝐢
3130sseli 3945 . . . . . . 7 (π‘₯ ∈ 𝐷 β†’ π‘₯ ∈ 𝐢)
3231ad2antrl 727 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ π‘₯ ∈ 𝐢)
3330sseli 3945 . . . . . . . 8 (𝑦 ∈ 𝐷 β†’ 𝑦 ∈ 𝐢)
3433adantl 483 . . . . . . 7 ((π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) β†’ 𝑦 ∈ 𝐢)
3534adantl 483 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ 𝑦 ∈ 𝐢)
3629, 22, 32, 35, 26ovmpod 7512 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯𝐽𝑦) = (π‘₯ RingHom 𝑦))
3719, 27, 363sstr4d 3996 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ (π‘₯ ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) β†’ (π‘₯𝐹𝑦) βŠ† (π‘₯𝐽𝑦))
3837ralrimivva 3198 . . 3 (π‘ˆ ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝐷 βˆ€π‘¦ ∈ 𝐷 (π‘₯𝐹𝑦) βŠ† (π‘₯𝐽𝑦))
39 ovex 7395 . . . . . 6 (π‘Ÿ RingHom 𝑠) ∈ V
409, 39fnmpoi 8007 . . . . 5 𝐹 Fn (𝐷 Γ— 𝐷)
4140a1i 11 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐹 Fn (𝐷 Γ— 𝐷))
4228, 39fnmpoi 8007 . . . . 5 𝐽 Fn (𝐢 Γ— 𝐢)
4342a1i 11 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐽 Fn (𝐢 Γ— 𝐢))
44 inex1g 5281 . . . . 5 (π‘ˆ ∈ 𝑉 β†’ (π‘ˆ ∩ DivRing) ∈ V)
4516, 44eqeltrid 2842 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ V)
4641, 43, 45isssc 17710 . . 3 (π‘ˆ ∈ 𝑉 β†’ (𝐹 βŠ†cat 𝐽 ↔ (𝐷 βŠ† 𝐢 ∧ βˆ€π‘₯ ∈ 𝐷 βˆ€π‘¦ ∈ 𝐷 (π‘₯𝐹𝑦) βŠ† (π‘₯𝐽𝑦))))
4718, 38, 46mpbir2and 712 . 2 (π‘ˆ ∈ 𝑉 β†’ 𝐹 βŠ†cat 𝐽)
4816, 28drhmsubc 46452 . . 3 (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)))
49 eqid 2737 . . . 4 ((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽) = ((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽)
5049subsubc 17746 . . 3 (𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)) β†’ (𝐹 ∈ (Subcatβ€˜((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽)) ↔ (𝐹 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)) ∧ 𝐹 βŠ†cat 𝐽)))
5148, 50syl 17 . 2 (π‘ˆ ∈ 𝑉 β†’ (𝐹 ∈ (Subcatβ€˜((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽)) ↔ (𝐹 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)) ∧ 𝐹 βŠ†cat 𝐽)))
5210, 47, 51mpbir2and 712 1 (π‘ˆ ∈ 𝑉 β†’ 𝐹 ∈ (Subcatβ€˜((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915   class class class wbr 5110   Γ— cxp 5636   Fn wfn 6496  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364   βŠ†cat cssc 17697   β†Ύcat cresc 17698  Subcatcsubc 17699  Ringcrg 19971  CRingccrg 19972   RingHom crh 20152  DivRingcdr 20199  Fieldcfield 20200  RingCatcringc 46375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-hom 17164  df-cco 17165  df-0g 17330  df-cat 17555  df-cid 17556  df-homf 17557  df-ssc 17700  df-resc 17701  df-subc 17702  df-estrc 18017  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-grp 18758  df-ghm 19013  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-rnghom 20155  df-drng 20201  df-field 20202  df-ringc 46377
This theorem is referenced by: (None)
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