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

Theorem rnghmsscmap 41235
Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)
Hypotheses
Ref Expression
rnghmsscmap.u (𝜑𝑈𝑉)
rnghmsscmap.r (𝜑𝑅 = (Rng ∩ 𝑈))
Assertion
Ref Expression
rnghmsscmap (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem rnghmsscmap
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnghmsscmap.r . . 3 (𝜑𝑅 = (Rng ∩ 𝑈))
2 inss2 3817 . . 3 (Rng ∩ 𝑈) ⊆ 𝑈
31, 2syl6eqss 3639 . 2 (𝜑𝑅𝑈)
4 eqid 2626 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
5 eqid 2626 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
64, 5rnghmf 41160 . . . . . 6 ( ∈ (𝑎 RngHomo 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
7 simpr 477 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
8 fvex 6160 . . . . . . . . . 10 (Base‘𝑏) ∈ V
9 fvex 6160 . . . . . . . . . 10 (Base‘𝑎) ∈ V
108, 9pm3.2i 471 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
11 elmapg 7816 . . . . . . . . 9 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
1210, 11mp1i 13 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
137, 12mpbird 247 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1413ex 450 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
156, 14syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RngHomo 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1615ssrdv 3594 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
17 ovres 6754 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
1817adantl 482 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
19 eqidd 2627 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
20 fveq2 6150 . . . . . . 7 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
21 fveq2 6150 . . . . . . 7 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2220, 21oveqan12rd 6625 . . . . . 6 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2322adantl 482 . . . . 5 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
243sseld 3587 . . . . . . . 8 (𝜑 → (𝑎𝑅𝑎𝑈))
2524com12 32 . . . . . . 7 (𝑎𝑅 → (𝜑𝑎𝑈))
2625adantr 481 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑎𝑈))
2726impcom 446 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑎𝑈)
283sseld 3587 . . . . . . . 8 (𝜑 → (𝑏𝑅𝑏𝑈))
2928com12 32 . . . . . . 7 (𝑏𝑅 → (𝜑𝑏𝑈))
3029adantl 482 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝜑𝑏𝑈))
3130impcom 446 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → 𝑏𝑈)
32 ovex 6633 . . . . . 6 ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V
3332a1i 11 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
3419, 23, 27, 31, 33ovmpt2d 6742 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
3516, 18, 343sstr4d 3632 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
3635ralrimivva 2970 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
37 rnghmfn 41151 . . . . 5 RngHomo Fn (Rng × Rng)
3837a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
39 inss1 3816 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
401, 39syl6eqss 3639 . . . . 5 (𝜑𝑅 ⊆ Rng)
41 xpss12 5191 . . . . 5 ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
4240, 40, 41syl2anc 692 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
43 fnssres 5964 . . . 4 (( RngHomo Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
4438, 42, 43syl2anc 692 . . 3 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
45 eqid 2626 . . . . 5 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
46 ovex 6633 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4745, 46fnmpt2i 7185 . . . 4 (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑈 × 𝑈)
4847a1i 11 . . 3 (𝜑 → (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑈 × 𝑈))
49 rnghmsscmap.u . . . 4 (𝜑𝑈𝑉)
50 elex 3203 . . . 4 (𝑈𝑉𝑈 ∈ V)
5149, 50syl 17 . . 3 (𝜑𝑈 ∈ V)
5244, 48, 51isssc 16396 . 2 (𝜑 → (( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↔ (𝑅𝑈 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))))
533, 36, 52mpbir2and 956 1 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  cin 3559  wss 3560   class class class wbr 4618   × cxp 5077  cres 5081   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑚 cmap 7803  Basecbs 15776  cat cssc 16383  Rngcrng 41135   RngHomo crngh 41146
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-ixp 7854  df-ssc 16386  df-ghm 17574  df-abl 18112  df-rng0 41136  df-rnghomo 41148
This theorem is referenced by:  rnghmsubcsetc  41238
  Copyright terms: Public domain W3C validator