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Theorem rhmsscmap2 43790
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
Hypotheses
Ref Expression
rhmsscmap.u (𝜑𝑈𝑉)
rhmsscmap.r (𝜑𝑅 = (Ring ∩ 𝑈))
Assertion
Ref Expression
rhmsscmap2 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Distinct variable group:   𝑥,𝑅,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rhmsscmap2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssidd 3917 . 2 (𝜑𝑅𝑅)
2 eqid 2797 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
3 eqid 2797 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
42, 3rhmf 19172 . . . . . 6 ( ∈ (𝑎 RingHom 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
5 simpr 485 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
6 fvex 6558 . . . . . . . . . 10 (Base‘𝑏) ∈ V
7 fvex 6558 . . . . . . . . . 10 (Base‘𝑎) ∈ V
86, 7pm3.2i 471 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
9 elmapg 8276 . . . . . . . . 9 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
108, 9mp1i 13 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
115, 10mpbird 258 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1211ex 413 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
134, 12syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RingHom 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1413ssrdv 3901 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
15 ovres 7177 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
1615adantl 482 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
17 eqidd 2798 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
18 fveq2 6545 . . . . . . . 8 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
19 fveq2 6545 . . . . . . . 8 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2018, 19oveqan12rd 7043 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2120adantl 482 . . . . . 6 (((𝑎𝑅𝑏𝑅) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
22 simpl 483 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑎𝑅)
23 simpr 485 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑏𝑅)
24 ovexd 7057 . . . . . 6 ((𝑎𝑅𝑏𝑅) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
2517, 21, 22, 23, 24ovmpod 7165 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2625adantl 482 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2714, 16, 263sstr4d 3941 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
2827ralrimivva 3160 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
29 rhmfn 43689 . . . . 5 RingHom Fn (Ring × Ring)
3029a1i 11 . . . 4 (𝜑 → RingHom Fn (Ring × Ring))
31 rhmsscmap.r . . . . . 6 (𝜑𝑅 = (Ring ∩ 𝑈))
32 inss1 4131 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
3331, 32syl6eqss 3948 . . . . 5 (𝜑𝑅 ⊆ Ring)
34 xpss12 5465 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
3533, 33, 34syl2anc 584 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
36 fnssres 6347 . . . 4 (( RingHom Fn (Ring × Ring) ∧ (𝑅 × 𝑅) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
3730, 35, 36syl2anc 584 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
38 eqid 2797 . . . . 5 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
39 ovex 7055 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4038, 39fnmpoi 7631 . . . 4 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅)
4140a1i 11 . . 3 (𝜑 → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅))
42 incom 4105 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
43 rhmsscmap.u . . . . . 6 (𝜑𝑈𝑉)
44 inex1g 5121 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4543, 44syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Ring) ∈ V)
4642, 45syl5eqel 2889 . . . 4 (𝜑 → (Ring ∩ 𝑈) ∈ V)
4731, 46eqeltrd 2885 . . 3 (𝜑𝑅 ∈ V)
4837, 41, 47isssc 16923 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↔ (𝑅𝑅 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))))
491, 28, 48mpbir2and 709 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wral 3107  Vcvv 3440  cin 3864  wss 3865   class class class wbr 4968   × cxp 5448  cres 5452   Fn wfn 6227  wf 6228  cfv 6232  (class class class)co 7023  cmpo 7025  𝑚 cmap 8263  Basecbs 16316  cat cssc 16910  Ringcrg 18991   RingHom crh 19158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-map 8265  df-ixp 8318  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-plusg 16411  df-0g 16548  df-ssc 16913  df-mhm 17778  df-ghm 18101  df-mgp 18934  df-ur 18946  df-ring 18993  df-rnghom 19161
This theorem is referenced by: (None)
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