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

Theorem rnghmval 42401
 Description: The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)
Hypotheses
Ref Expression
isrnghm.b 𝐵 = (Base‘𝑅)
isrnghm.t · = (.r𝑅)
isrnghm.m = (.r𝑆)
rnghmval.c 𝐶 = (Base‘𝑆)
rnghmval.p + = (+g𝑅)
rnghmval.a = (+g𝑆)
Assertion
Ref Expression
rnghmval ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
Distinct variable groups:   𝐵,𝑓,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝐶,𝑓
Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦,𝑓)   (𝑥,𝑦,𝑓)   · (𝑥,𝑦,𝑓)   (𝑥,𝑦,𝑓)

Proof of Theorem rnghmval
Dummy variables 𝑟 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rnghomo 42397 . . 3 RngHomo = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
21a1i 11 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → RngHomo = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))}))
3 fveq2 6352 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 isrnghm.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2812 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
65csbeq1d 3681 . . . . 5 (𝑟 = 𝑅(Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
7 fveq2 6352 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
8 rnghmval.c . . . . . . . 8 𝐶 = (Base‘𝑆)
97, 8syl6eqr 2812 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐶)
109csbeq1d 3681 . . . . . 6 (𝑠 = 𝑆(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
1110csbeq2dv 4135 . . . . 5 (𝑠 = 𝑆𝐵 / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
126, 11sylan9eq 2814 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
1312adantl 473 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
14 fvex 6362 . . . . . . . 8 (Base‘𝑅) ∈ V
154, 14eqeltri 2835 . . . . . . 7 𝐵 ∈ V
16 fvex 6362 . . . . . . . 8 (Base‘𝑆) ∈ V
178, 16eqeltri 2835 . . . . . . 7 𝐶 ∈ V
18 oveq12 6822 . . . . . . . . 9 ((𝑤 = 𝐶𝑣 = 𝐵) → (𝑤𝑚 𝑣) = (𝐶𝑚 𝐵))
1918ancoms 468 . . . . . . . 8 ((𝑣 = 𝐵𝑤 = 𝐶) → (𝑤𝑚 𝑣) = (𝐶𝑚 𝐵))
20 raleq 3277 . . . . . . . . . 10 (𝑣 = 𝐵 → (∀𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
2120raleqbi1dv 3285 . . . . . . . . 9 (𝑣 = 𝐵 → (∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
2221adantr 472 . . . . . . . 8 ((𝑣 = 𝐵𝑤 = 𝐶) → (∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
2319, 22rabeqbidv 3335 . . . . . . 7 ((𝑣 = 𝐵𝑤 = 𝐶) → {𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
2415, 17, 23csbie2 3704 . . . . . 6 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))}
2524a1i 11 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
26 fveq2 6352 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (+g𝑟) = (+g𝑅))
27 rnghmval.p . . . . . . . . . . . . 13 + = (+g𝑅)
2826, 27syl6eqr 2812 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (+g𝑟) = + )
2928oveqdr 6837 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(+g𝑟)𝑦) = (𝑥 + 𝑦))
3029fveq2d 6356 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(+g𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
31 fveq2 6352 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
32 rnghmval.a . . . . . . . . . . . . 13 = (+g𝑆)
3331, 32syl6eqr 2812 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = )
3433adantl 473 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑠 = 𝑆) → (+g𝑠) = )
3534oveqd 6830 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
3630, 35eqeq12d 2775 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
37 fveq2 6352 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
38 isrnghm.t . . . . . . . . . . . . 13 · = (.r𝑅)
3937, 38syl6eqr 2812 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = · )
4039oveqdr 6837 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
4140fveq2d 6356 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(.r𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
42 fveq2 6352 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (.r𝑠) = (.r𝑆))
43 isrnghm.m . . . . . . . . . . . . 13 = (.r𝑆)
4442, 43syl6eqr 2812 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (.r𝑠) = )
4544adantl 473 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑠 = 𝑆) → (.r𝑠) = )
4645oveqd 6830 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
4741, 46eqeq12d 2775 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
4836, 47anbi12d 749 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))))
4948ralbidv 3124 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (∀𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))))
5049ralbidv 3124 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))))
5150rabbidv 3329 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
5225, 51eqtrd 2794 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
5352adantl 473 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
5413, 53eqtrd 2794 . 2 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
55 simpl 474 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑅 ∈ Rng)
56 simpr 479 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑆 ∈ Rng)
57 ovex 6841 . . . 4 (𝐶𝑚 𝐵) ∈ V
5857rabex 4964 . . 3 {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))} ∈ V
5958a1i 11 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))} ∈ V)
602, 54, 55, 56, 59ovmpt2d 6953 1 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  Vcvv 3340  ⦋csb 3674  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815   ↑𝑚 cmap 8023  Basecbs 16059  +gcplusg 16143  .rcmulr 16144  Rngcrng 42384   RngHomo crngh 42395 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-rnghomo 42397 This theorem is referenced by:  isrnghm  42402
 Copyright terms: Public domain W3C validator