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Theorem rngcifuestrc 44100
Description: The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.)
Hypotheses
Ref Expression
rngcifuestrc.r 𝑅 = (RngCat‘𝑈)
rngcifuestrc.e 𝐸 = (ExtStrCat‘𝑈)
rngcifuestrc.b 𝐵 = (Base‘𝑅)
rngcifuestrc.u (𝜑𝑈𝑉)
rngcifuestrc.f (𝜑𝐹 = ( I ↾ 𝐵))
rngcifuestrc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
Assertion
Ref Expression
rngcifuestrc (𝜑𝐹(𝑅 Func 𝐸)𝐺)
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rngcifuestrc
StepHypRef Expression
1 eqid 2826 . . . . 5 (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2 rngcifuestrc.u . . . . 5 (𝜑𝑈𝑉)
3 rngcifuestrc.r . . . . . . 7 𝑅 = (RngCat‘𝑈)
4 rngcifuestrc.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4, 2rngcbas 44068 . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Rng))
6 incom 4182 . . . . . 6 (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
75, 6syl6eq 2877 . . . . 5 (𝜑𝐵 = (Rng ∩ 𝑈))
8 eqid 2826 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
93, 4, 2, 8rngchomfval 44069 . . . . 5 (𝜑 → (Hom ‘𝑅) = ( RngHomo ↾ (𝐵 × 𝐵)))
101, 2, 7, 9rnghmsubcsetc 44080 . . . 4 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
1110idi 1 . . 3 (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
12 eqid 2826 . . 3 ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))
13 eqid 2826 . . 3 (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
14 rngcifuestrc.f . . . 4 (𝜑𝐹 = ( I ↾ 𝐵))
153, 2, 5, 9rngcval 44065 . . . . . . 7 (𝜑𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
1615fveq2d 6671 . . . . . 6 (𝜑 → (Base‘𝑅) = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
174, 16syl5eq 2873 . . . . 5 (𝜑𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
1817reseq2d 5852 . . . 4 (𝜑 → ( I ↾ 𝐵) = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
1914, 18eqtrd 2861 . . 3 (𝜑𝐹 = ( I ↾ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))))
20 rngcifuestrc.g . . . 4 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
2117adantr 481 . . . . 5 ((𝜑𝑥𝐵) → 𝐵 = (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))))
229oveqdr 7176 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(Hom ‘𝑅)𝑦) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦))
23 ovres 7304 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
2423adantl 482 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑦) = (𝑥 RngHomo 𝑦))
2522, 24eqtr2d 2862 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 RngHomo 𝑦) = (𝑥(Hom ‘𝑅)𝑦))
2625reseq2d 5852 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥(Hom ‘𝑅)𝑦)))
2717, 21, 26mpoeq123dva 7220 . . . 4 (𝜑 → (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
2820, 27eqtrd 2861 . . 3 (𝜑𝐺 = (𝑥 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))), 𝑦 ∈ (Base‘((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅))) ↦ ( I ↾ (𝑥(Hom ‘𝑅)𝑦))))
2911, 12, 13, 19, 28inclfusubc 43970 . 2 (𝜑𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺)
30 rngcifuestrc.e . . . . 5 𝐸 = (ExtStrCat‘𝑈)
3130a1i 11 . . . 4 (𝜑𝐸 = (ExtStrCat‘𝑈))
3215, 31oveq12d 7166 . . 3 (𝜑 → (𝑅 Func 𝐸) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈)))
3332breqd 5074 . 2 (𝜑 → (𝐹(𝑅 Func 𝐸)𝐺𝐹(((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func (ExtStrCat‘𝑈))𝐺))
3429, 33mpbird 258 1 (𝜑𝐹(𝑅 Func 𝐸)𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  cin 3939   class class class wbr 5063   I cid 5458   × cxp 5552  cres 5556  cfv 6352  (class class class)co 7148  cmpo 7150  Basecbs 16473  Hom chom 16566  cat cresc 17068  Subcatcsubc 17069   Func cfunc 17114  ExtStrCatcestrc 17362  Rngcrng 43977   RngHomo crngh 43988  RngCatcrngc 44060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-hom 16579  df-cco 16580  df-0g 16705  df-cat 16929  df-cid 16930  df-homf 16931  df-ssc 17070  df-resc 17071  df-subc 17072  df-func 17118  df-idfu 17119  df-full 17164  df-fth 17165  df-estrc 17363  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-mhm 17944  df-grp 18036  df-ghm 18286  df-abl 18829  df-mgp 19160  df-mgmhm 43878  df-rng0 43978  df-rnghomo 43990  df-rngc 44062
This theorem is referenced by:  funcrngcsetcALT  44102
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