Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcringcsetclem8ALTV Structured version   Visualization version   GIF version

Theorem funcringcsetclem8ALTV 45057
 Description: Lemma 8 for funcringcsetcALTV 45059. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTV‘𝑈)
funcringcsetcALTV.s 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV.b 𝐵 = (Base‘𝑅)
funcringcsetcALTV.c 𝐶 = (Base‘𝑆)
funcringcsetcALTV.u (𝜑𝑈 ∈ WUni)
funcringcsetcALTV.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem8ALTV ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcringcsetclem8ALTV
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1oi 6639 . . . 4 ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)–1-1-onto→(𝑋 RingHom 𝑌)
2 f1of 6602 . . . 4 (( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)–1-1-onto→(𝑋 RingHom 𝑌) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶(𝑋 RingHom 𝑌))
31, 2mp1i 13 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶(𝑋 RingHom 𝑌))
4 eqid 2758 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
5 eqid 2758 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
64, 5rhmf 19549 . . . . 5 (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
7 fvex 6671 . . . . . . . . . 10 (Base‘𝑌) ∈ V
8 fvex 6671 . . . . . . . . . 10 (Base‘𝑋) ∈ V
97, 8pm3.2i 474 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
10 elmapg 8429 . . . . . . . . . 10 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
1110bicomd 226 . . . . . . . . 9 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
129, 11mp1i 13 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
1312biimpa 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
14 simpr 488 . . . . . . . . . 10 ((𝑋𝐵𝑌𝐵) → 𝑌𝐵)
15 funcringcsetcALTV.r . . . . . . . . . . 11 𝑅 = (RingCatALTV‘𝑈)
16 funcringcsetcALTV.s . . . . . . . . . . 11 𝑆 = (SetCat‘𝑈)
17 funcringcsetcALTV.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
18 funcringcsetcALTV.c . . . . . . . . . . 11 𝐶 = (Base‘𝑆)
19 funcringcsetcALTV.u . . . . . . . . . . 11 (𝜑𝑈 ∈ WUni)
20 funcringcsetcALTV.f . . . . . . . . . . 11 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
2115, 16, 17, 18, 19, 20funcringcsetclem1ALTV 45050 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
2214, 21sylan2 595 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
23 simpl 486 . . . . . . . . . 10 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
2415, 16, 17, 18, 19, 20funcringcsetclem1ALTV 45050 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2523, 24sylan2 595 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2622, 25oveq12d 7168 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2726adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2813, 27eleqtrrd 2855 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋)))
2928ex 416 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋))))
306, 29syl5 34 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋))))
3130ssrdv 3898 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 RingHom 𝑌) ⊆ ((𝐹𝑌) ↑m (𝐹𝑋)))
323, 31fssd 6513 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶((𝐹𝑌) ↑m (𝐹𝑋)))
33 funcringcsetcALTV.g . . . 4 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
3415, 16, 17, 18, 19, 20, 33funcringcsetclem5ALTV 45054 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
3519adantr 484 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
36 eqid 2758 . . . 4 (Hom ‘𝑅) = (Hom ‘𝑅)
3723adantl 485 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3814adantl 485 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
3915, 17, 35, 36, 37, 38ringchomALTV 45039 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
40 eqid 2758 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4115, 16, 17, 18, 19, 20funcringcsetclem2ALTV 45051 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4223, 41sylan2 595 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4315, 16, 17, 18, 19, 20funcringcsetclem2ALTV 45051 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
4414, 43sylan2 595 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
4516, 35, 40, 42, 44setchom 17406 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑m (𝐹𝑋)))
4634, 39, 45feq123d 6487 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) ↔ ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶((𝐹𝑌) ↑m (𝐹𝑋))))
4732, 46mpbird 260 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ↦ cmpt 5112   I cid 5429   ↾ cres 5526  ⟶wf 6331  –1-1-onto→wf1o 6334  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416  WUnicwun 10160  Basecbs 16541  Hom chom 16634  SetCatcsetc 17401   RingHom crh 19535  RingCatALTVcringcALTV 44995 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-wun 10162  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-z 12021  df-dec 12138  df-uz 12283  df-fz 12940  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-plusg 16636  df-hom 16647  df-cco 16648  df-0g 16773  df-setc 17402  df-mhm 18022  df-ghm 18423  df-mgp 19308  df-ur 19320  df-ring 19367  df-rnghom 19538  df-ringcALTV 44997 This theorem is referenced by:  funcringcsetcALTV  45059
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