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Theorem fldhmsubcALTV 41420
Description: According to df-subc 16412, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16440 and subcss2 16443). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
drhmsubcALTV.c 𝐶 = (𝑈 ∩ DivRing)
drhmsubcALTV.j 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
fldhmsubcALTV.d 𝐷 = (𝑈 ∩ Field)
fldhmsubcALTV.f 𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))
Assertion
Ref Expression
fldhmsubcALTV (𝑈𝑉𝐹 ∈ (Subcat‘((RingCatALTV‘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 fldhmsubcALTV
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3780 . . . . . . 7 (𝑟 ∈ (DivRing ∩ CRing) ↔ (𝑟 ∈ DivRing ∧ 𝑟 ∈ CRing))
21simprbi 480 . . . . . 6 (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ CRing)
3 crngring 18498 . . . . . 6 (𝑟 ∈ CRing → 𝑟 ∈ Ring)
42, 3syl 17 . . . . 5 (𝑟 ∈ (DivRing ∩ CRing) → 𝑟 ∈ Ring)
5 df-field 18690 . . . . 5 Field = (DivRing ∩ CRing)
64, 5eleq2s 2716 . . . 4 (𝑟 ∈ Field → 𝑟 ∈ Ring)
76rgen 2918 . . 3 𝑟 ∈ Field 𝑟 ∈ Ring
8 fldhmsubcALTV.d . . 3 𝐷 = (𝑈 ∩ Field)
9 fldhmsubcALTV.f . . 3 𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))
107, 8, 9srhmsubcALTV 41412 . 2 (𝑈𝑉𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)))
11 inss1 3817 . . . . . . 7 (DivRing ∩ CRing) ⊆ DivRing
125, 11eqsstri 3620 . . . . . 6 Field ⊆ DivRing
13 sslin 3823 . . . . . 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 drhmsubcALTV.c . . . . 5 𝐶 = (𝑈 ∩ DivRing)
178, 16sseq12i 3616 . . . 4 (𝐷𝐶 ↔ (𝑈 ∩ Field) ⊆ (𝑈 ∩ DivRing))
1815, 17sylibr 224 . . 3 (𝑈𝑉𝐷𝐶)
19 ssid 3609 . . . . . 6 (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦)
2019a1i 11 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥 RingHom 𝑦) ⊆ (𝑥 RingHom 𝑦))
219a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠)))
22 oveq12 6624 . . . . . . 7 ((𝑟 = 𝑥𝑠 = 𝑦) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
2322adantl 482 . . . . . 6 (((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) ∧ (𝑟 = 𝑥𝑠 = 𝑦)) → (𝑟 RingHom 𝑠) = (𝑥 RingHom 𝑦))
24 simprl 793 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑥𝐷)
25 simpr 477 . . . . . . 7 ((𝑥𝐷𝑦𝐷) → 𝑦𝐷)
2625adantl 482 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑦𝐷)
27 ovexd 6645 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥 RingHom 𝑦) ∈ V)
2821, 23, 24, 26, 27ovmpt2d 6753 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐹𝑦) = (𝑥 RingHom 𝑦))
29 drhmsubcALTV.j . . . . . . 7 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))
3029a1i 11 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠)))
3114, 17mpbir 221 . . . . . . . 8 𝐷𝐶
3231sseli 3584 . . . . . . 7 (𝑥𝐷𝑥𝐶)
3332ad2antrl 763 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑥𝐶)
3431sseli 3584 . . . . . . . 8 (𝑦𝐷𝑦𝐶)
3534adantl 482 . . . . . . 7 ((𝑥𝐷𝑦𝐷) → 𝑦𝐶)
3635adantl 482 . . . . . 6 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → 𝑦𝐶)
3730, 23, 33, 36, 27ovmpt2d 6753 . . . . 5 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐽𝑦) = (𝑥 RingHom 𝑦))
3820, 28, 373sstr4d 3633 . . . 4 ((𝑈𝑉 ∧ (𝑥𝐷𝑦𝐷)) → (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
3938ralrimivva 2967 . . 3 (𝑈𝑉 → ∀𝑥𝐷𝑦𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))
40 ovex 6643 . . . . . 6 (𝑟 RingHom 𝑠) ∈ V
419, 40fnmpt2i 7199 . . . . 5 𝐹 Fn (𝐷 × 𝐷)
4241a1i 11 . . . 4 (𝑈𝑉𝐹 Fn (𝐷 × 𝐷))
4329, 40fnmpt2i 7199 . . . . 5 𝐽 Fn (𝐶 × 𝐶)
4443a1i 11 . . . 4 (𝑈𝑉𝐽 Fn (𝐶 × 𝐶))
45 inex1g 4771 . . . . 5 (𝑈𝑉 → (𝑈 ∩ DivRing) ∈ V)
4616, 45syl5eqel 2702 . . . 4 (𝑈𝑉𝐶 ∈ V)
4742, 44, 46isssc 16420 . . 3 (𝑈𝑉 → (𝐹cat 𝐽 ↔ (𝐷𝐶 ∧ ∀𝑥𝐷𝑦𝐷 (𝑥𝐹𝑦) ⊆ (𝑥𝐽𝑦))))
4818, 39, 47mpbir2and 956 . 2 (𝑈𝑉𝐹cat 𝐽)
4916, 29drhmsubcALTV 41416 . . 3 (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
50 eqid 2621 . . . 4 ((RingCatALTV‘𝑈) ↾cat 𝐽) = ((RingCatALTV‘𝑈) ↾cat 𝐽)
5150subsubc 16453 . . 3 (𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)) → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹cat 𝐽)))
5249, 51syl 17 . 2 (𝑈𝑉 → (𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)) ↔ (𝐹 ∈ (Subcat‘(RingCatALTV‘𝑈)) ∧ 𝐹cat 𝐽)))
5310, 48, 52mpbir2and 956 1 (𝑈𝑉𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cin 3559  wss 3560   class class class wbr 4623   × cxp 5082   Fn wfn 5852  cfv 5857  (class class class)co 6615  cmpt2 6617  cat cssc 16407  cat cresc 16408  Subcatcsubc 16409  Ringcrg 18487  CRingccrg 18488   RingHom crh 18652  DivRingcdr 18687  Fieldcfield 18688  RingCatALTVcringcALTV 41322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-hom 15906  df-cco 15907  df-0g 16042  df-cat 16269  df-cid 16270  df-homf 16271  df-ssc 16410  df-resc 16411  df-subc 16412  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-grp 17365  df-ghm 17598  df-mgp 18430  df-ur 18442  df-ring 18489  df-cring 18490  df-rnghom 18655  df-drng 18689  df-field 18690  df-ringcALTV 41324
This theorem is referenced by: (None)
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