Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnghmsscmap2 Structured version   Visualization version   GIF version

Theorem rnghmsscmap2 41765
 Description: The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
Hypotheses
Ref Expression
rnghmsscmap.u (𝜑𝑈𝑉)
rnghmsscmap.r (𝜑𝑅 = (Rng ∩ 𝑈))
Assertion
Ref Expression
rnghmsscmap2 (𝜑 → ( RngHomo ↾ (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 rnghmsscmap2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3587 . . 3 𝑅𝑅
21a1i 11 . 2 (𝜑𝑅𝑅)
3 eqid 2610 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
4 eqid 2610 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
53, 4rnghmf 41689 . . . . . 6 ( ∈ (𝑎 RngHomo 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
6 simpr 476 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
7 fvex 6113 . . . . . . . . . 10 (Base‘𝑏) ∈ V
8 fvex 6113 . . . . . . . . . 10 (Base‘𝑎) ∈ V
97, 8pm3.2i 470 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
10 elmapg 7757 . . . . . . . . 9 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
119, 10mp1i 13 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
126, 11mpbird 246 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1312ex 449 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
145, 13syl5 33 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RngHomo 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1514ssrdv 3574 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RngHomo 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
16 ovres 6698 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
1716adantl 481 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RngHomo 𝑏))
18 eqidd 2611 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
19 fveq2 6103 . . . . . . . 8 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
20 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2119, 20oveqan12rd 6569 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2221adantl 481 . . . . . 6 (((𝑎𝑅𝑏𝑅) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
23 simpl 472 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑎𝑅)
24 simpr 476 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑏𝑅)
25 ovex 6577 . . . . . . 7 ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V
2625a1i 11 . . . . . 6 ((𝑎𝑅𝑏𝑅) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
2718, 22, 23, 24, 26ovmpt2d 6686 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2827adantl 481 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2915, 17, 283sstr4d 3611 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
3029ralrimivva 2954 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
31 rnghmfn 41680 . . . . 5 RngHomo Fn (Rng × Rng)
3231a1i 11 . . . 4 (𝜑 → RngHomo Fn (Rng × Rng))
33 rnghmsscmap.r . . . . . 6 (𝜑𝑅 = (Rng ∩ 𝑈))
34 inss1 3795 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
3533, 34syl6eqss 3618 . . . . 5 (𝜑𝑅 ⊆ Rng)
36 xpss12 5148 . . . . 5 ((𝑅 ⊆ Rng ∧ 𝑅 ⊆ Rng) → (𝑅 × 𝑅) ⊆ (Rng × Rng))
3735, 35, 36syl2anc 691 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Rng × Rng))
38 fnssres 5918 . . . 4 (( RngHomo Fn (Rng × Rng) ∧ (𝑅 × 𝑅) ⊆ (Rng × Rng)) → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
3932, 37, 38syl2anc 691 . . 3 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
40 eqid 2610 . . . . 5 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
41 ovex 6577 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4240, 41fnmpt2i 7128 . . . 4 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅)
4342a1i 11 . . 3 (𝜑 → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅))
44 incom 3767 . . . . 5 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
45 rnghmsscmap.u . . . . . 6 (𝜑𝑈𝑉)
46 inex1g 4729 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4745, 46syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Rng) ∈ V)
4844, 47syl5eqel 2692 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
4933, 48eqeltrd 2688 . . 3 (𝜑𝑅 ∈ V)
5039, 43, 49isssc 16303 . 2 (𝜑 → (( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↔ (𝑅𝑅 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RngHomo ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))))
512, 30, 50mpbir2and 959 1 (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   × cxp 5036   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  Basecbs 15695   ⊆cat cssc 16290  Rngcrng 41664   RngHomo crngh 41675 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-ssc 16293  df-ghm 17481  df-abl 18019  df-rng0 41665  df-rnghomo 41677 This theorem is referenced by: (None)
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