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Theorem funcringcsetcALTV2lem9 42572
Description: Lemma 9 for funcringcsetcALTV2 42573. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV2.r 𝑅 = (RingCat‘𝑈)
funcringcsetcALTV2.s 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV2.b 𝐵 = (Base‘𝑅)
funcringcsetcALTV2.c 𝐶 = (Base‘𝑆)
funcringcsetcALTV2.u (𝜑𝑈 ∈ WUni)
funcringcsetcALTV2.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV2.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetcALTV2lem9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦   𝑥,𝑍,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem funcringcsetcALTV2lem9
StepHypRef Expression
1 funcringcsetcALTV2.r . . . . . 6 𝑅 = (RingCat‘𝑈)
2 funcringcsetcALTV2.b . . . . . 6 𝐵 = (Base‘𝑅)
3 funcringcsetcALTV2.u . . . . . . 7 (𝜑𝑈 ∈ WUni)
43adantr 466 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑈 ∈ WUni)
5 eqid 2771 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
6 simpr1 1233 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
7 simpr2 1235 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
81, 2, 4, 5, 6, 7ringchom 42541 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
98eleq2d 2836 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ↔ 𝐻 ∈ (𝑋 RingHom 𝑌)))
10 simpr3 1237 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
111, 2, 4, 5, 7, 10ringchom 42541 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌(Hom ‘𝑅)𝑍) = (𝑌 RingHom 𝑍))
1211eleq2d 2836 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍) ↔ 𝐾 ∈ (𝑌 RingHom 𝑍)))
139, 12anbi12d 616 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍)) ↔ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))))
14 rhmco 18947 . . . . . . . 8 ((𝐾 ∈ (𝑌 RingHom 𝑍) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → (𝐾𝐻) ∈ (𝑋 RingHom 𝑍))
1514ancoms 455 . . . . . . 7 ((𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → (𝐾𝐻) ∈ (𝑋 RingHom 𝑍))
1615adantl 467 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾𝐻) ∈ (𝑋 RingHom 𝑍))
17 fvresi 6583 . . . . . 6 ((𝐾𝐻) ∈ (𝑋 RingHom 𝑍) → (( I ↾ (𝑋 RingHom 𝑍))‘(𝐾𝐻)) = (𝐾𝐻))
1816, 17syl 17 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (( I ↾ (𝑋 RingHom 𝑍))‘(𝐾𝐻)) = (𝐾𝐻))
19 funcringcsetcALTV2.s . . . . . . . . 9 𝑆 = (SetCat‘𝑈)
20 funcringcsetcALTV2.c . . . . . . . . 9 𝐶 = (Base‘𝑆)
21 funcringcsetcALTV2.f . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
22 funcringcsetcALTV2.g . . . . . . . . 9 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
231, 19, 2, 20, 3, 21, 22funcringcsetcALTV2lem5 42568 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
24233adantr2 1175 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
2524adantr 466 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
264adantr 466 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑈 ∈ WUni)
27 eqid 2771 . . . . . . 7 (comp‘𝑅) = (comp‘𝑅)
281, 2, 3ringcbas 42539 . . . . . . . . . . . . 13 (𝜑𝐵 = (𝑈 ∩ Ring))
29 inss1 3981 . . . . . . . . . . . . 13 (𝑈 ∩ Ring) ⊆ 𝑈
3028, 29syl6eqss 3804 . . . . . . . . . . . 12 (𝜑𝐵𝑈)
3130sseld 3751 . . . . . . . . . . 11 (𝜑 → (𝑋𝐵𝑋𝑈))
3231com12 32 . . . . . . . . . 10 (𝑋𝐵 → (𝜑𝑋𝑈))
33323ad2ant1 1127 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑋𝑈))
3433impcom 394 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝑈)
3534adantr 466 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑋𝑈)
3630sseld 3751 . . . . . . . . . . 11 (𝜑 → (𝑌𝐵𝑌𝑈))
3736com12 32 . . . . . . . . . 10 (𝑌𝐵 → (𝜑𝑌𝑈))
38373ad2ant2 1128 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑌𝑈))
3938impcom 394 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝑈)
4039adantr 466 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑌𝑈)
4130sseld 3751 . . . . . . . . . . 11 (𝜑 → (𝑍𝐵𝑍𝑈))
4241com12 32 . . . . . . . . . 10 (𝑍𝐵 → (𝜑𝑍𝑈))
43423ad2ant3 1129 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑍𝑈))
4443impcom 394 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝑈)
4544adantr 466 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑍𝑈)
46 eqid 2771 . . . . . . . . 9 (Base‘𝑋) = (Base‘𝑋)
47 eqid 2771 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
4846, 47rhmf 18936 . . . . . . . 8 (𝐻 ∈ (𝑋 RingHom 𝑌) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
4948ad2antrl 707 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
50 eqid 2771 . . . . . . . . 9 (Base‘𝑍) = (Base‘𝑍)
5147, 50rhmf 18936 . . . . . . . 8 (𝐾 ∈ (𝑌 RingHom 𝑍) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
5251ad2antll 708 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
531, 26, 27, 35, 40, 45, 49, 52ringcco 42545 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻) = (𝐾𝐻))
5425, 53fveq12d 6338 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (( I ↾ (𝑋 RingHom 𝑍))‘(𝐾𝐻)))
55 eqid 2771 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
561, 19, 2, 20, 3, 21funcringcsetcALTV2lem2 42565 . . . . . . . . 9 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
57563ad2antr1 1203 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) ∈ 𝑈)
5857adantr 466 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹𝑋) ∈ 𝑈)
591, 19, 2, 20, 3, 21funcringcsetcALTV2lem2 42565 . . . . . . . . 9 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
60593ad2antr2 1204 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) ∈ 𝑈)
6160adantr 466 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹𝑌) ∈ 𝑈)
621, 19, 2, 20, 3, 21funcringcsetcALTV2lem2 42565 . . . . . . . . 9 ((𝜑𝑍𝐵) → (𝐹𝑍) ∈ 𝑈)
63623ad2antr3 1205 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) ∈ 𝑈)
6463adantr 466 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹𝑍) ∈ 𝑈)
651, 19, 2, 20, 3, 21funcringcsetcALTV2lem1 42564 . . . . . . . . . . . 12 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
66653ad2antr1 1203 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) = (Base‘𝑋))
671, 19, 2, 20, 3, 21funcringcsetcALTV2lem1 42564 . . . . . . . . . . . 12 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
68673ad2antr2 1204 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) = (Base‘𝑌))
6966, 68feq23d 6180 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7069adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7149, 70mpbird 247 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻:(𝐹𝑋)⟶(𝐹𝑌))
72 simpll 750 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝜑)
73 3simpa 1142 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋𝐵𝑌𝐵))
7473ad2antlr 706 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑋𝐵𝑌𝐵))
75 simprl 754 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻 ∈ (𝑋 RingHom 𝑌))
761, 19, 2, 20, 3, 21, 22funcringcsetcALTV2lem6 42569 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7772, 74, 75, 76syl3anc 1476 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7877feq1d 6170 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(𝐹𝑋)⟶(𝐹𝑌)))
7971, 78mpbird 247 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌))
801, 19, 2, 20, 3, 21funcringcsetcALTV2lem1 42564 . . . . . . . . . . . 12 ((𝜑𝑍𝐵) → (𝐹𝑍) = (Base‘𝑍))
81803ad2antr3 1205 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) = (Base‘𝑍))
8268, 81feq23d 6180 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8382adantr 466 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8452, 83mpbird 247 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾:(𝐹𝑌)⟶(𝐹𝑍))
85 3simpc 1146 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌𝐵𝑍𝐵))
8685ad2antlr 706 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑌𝐵𝑍𝐵))
87 simprr 756 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾 ∈ (𝑌 RingHom 𝑍))
881, 19, 2, 20, 3, 21, 22funcringcsetcALTV2lem6 42569 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐵𝑍𝐵) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
8972, 86, 87, 88syl3anc 1476 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9089feq1d 6170 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(𝐹𝑌)⟶(𝐹𝑍)))
9184, 90mpbird 247 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍))
9219, 26, 55, 58, 61, 64, 79, 91setcco 16940 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
9389, 77coeq12d 5425 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9492, 93eqtrd 2805 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9518, 54, 943eqtr4d 2815 . . . 4 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
9695ex 397 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
9713, 96sylbid 230 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
98973impia 1109 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  cin 3722  cop 4322  cmpt 4863   I cid 5156  cres 5251  ccom 5253  wf 6027  cfv 6031  (class class class)co 6793  cmpt2 6795  WUnicwun 9724  Basecbs 16064  Hom chom 16160  compcco 16161  SetCatcsetc 16932  Ringcrg 18755   RingHom crh 18922  RingCatcringc 42531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-wun 9726  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-uz 11889  df-fz 12534  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-hom 16174  df-cco 16175  df-0g 16310  df-resc 16678  df-setc 16933  df-estrc 16970  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-grp 17633  df-ghm 17866  df-mgp 18698  df-ur 18710  df-ring 18757  df-rnghom 18925  df-ringc 42533
This theorem is referenced by:  funcringcsetcALTV2  42573
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