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Theorem funcringcsetcALTV2lem9 46654
Description: Lemma 9 for funcringcsetcALTV2 46655. (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 481 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑈 ∈ WUni)
5 eqid 2732 . . . . . 6 (Hom ‘𝑅) = (Hom ‘𝑅)
6 simpr1 1194 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
7 simpr2 1195 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
81, 2, 4, 5, 6, 7ringchom 46623 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
98eleq2d 2819 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ↔ 𝐻 ∈ (𝑋 RingHom 𝑌)))
10 simpr3 1196 . . . . . 6 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
111, 2, 4, 5, 7, 10ringchom 46623 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌(Hom ‘𝑅)𝑍) = (𝑌 RingHom 𝑍))
1211eleq2d 2819 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍) ↔ 𝐾 ∈ (𝑌 RingHom 𝑍)))
139, 12anbi12d 631 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍)) ↔ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))))
14 rhmco 20228 . . . . . . . 8 ((𝐾 ∈ (𝑌 RingHom 𝑍) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → (𝐾𝐻) ∈ (𝑋 RingHom 𝑍))
1514ancoms 459 . . . . . . 7 ((𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → (𝐾𝐻) ∈ (𝑋 RingHom 𝑍))
1615adantl 482 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾𝐻) ∈ (𝑋 RingHom 𝑍))
17 fvresi 7156 . . . . . 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 46650 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
24233adantr2 1170 . . . . . . 7 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
2524adantr 481 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑋𝐺𝑍) = ( I ↾ (𝑋 RingHom 𝑍)))
264adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑈 ∈ WUni)
27 eqid 2732 . . . . . . 7 (comp‘𝑅) = (comp‘𝑅)
281, 2, 3ringcbas 46621 . . . . . . . . . . . . 13 (𝜑𝐵 = (𝑈 ∩ Ring))
29 inss1 4225 . . . . . . . . . . . . 13 (𝑈 ∩ Ring) ⊆ 𝑈
3028, 29eqsstrdi 4033 . . . . . . . . . . . 12 (𝜑𝐵𝑈)
3130sseld 3978 . . . . . . . . . . 11 (𝜑 → (𝑋𝐵𝑋𝑈))
3231com12 32 . . . . . . . . . 10 (𝑋𝐵 → (𝜑𝑋𝑈))
33323ad2ant1 1133 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑋𝑈))
3433impcom 408 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝑈)
3534adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑋𝑈)
3630sseld 3978 . . . . . . . . . . 11 (𝜑 → (𝑌𝐵𝑌𝑈))
3736com12 32 . . . . . . . . . 10 (𝑌𝐵 → (𝜑𝑌𝑈))
38373ad2ant2 1134 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑌𝑈))
3938impcom 408 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝑈)
4039adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑌𝑈)
4130sseld 3978 . . . . . . . . . . 11 (𝜑 → (𝑍𝐵𝑍𝑈))
4241com12 32 . . . . . . . . . 10 (𝑍𝐵 → (𝜑𝑍𝑈))
43423ad2ant3 1135 . . . . . . . . 9 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝜑𝑍𝑈))
4443impcom 408 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝑈)
4544adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝑍𝑈)
46 eqid 2732 . . . . . . . . 9 (Base‘𝑋) = (Base‘𝑋)
47 eqid 2732 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
4846, 47rhmf 20215 . . . . . . . 8 (𝐻 ∈ (𝑋 RingHom 𝑌) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
4948ad2antrl 726 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻:(Base‘𝑋)⟶(Base‘𝑌))
50 eqid 2732 . . . . . . . . 9 (Base‘𝑍) = (Base‘𝑍)
5147, 50rhmf 20215 . . . . . . . 8 (𝐾 ∈ (𝑌 RingHom 𝑍) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
5251ad2antll 727 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾:(Base‘𝑌)⟶(Base‘𝑍))
531, 26, 27, 35, 40, 45, 49, 52ringcco 46627 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻) = (𝐾𝐻))
5425, 53fveq12d 6886 . . . . 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 2732 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
561, 19, 2, 20, 3, 21funcringcsetcALTV2lem2 46647 . . . . . . . . 9 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
57563ad2antr1 1188 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) ∈ 𝑈)
5857adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹𝑋) ∈ 𝑈)
591, 19, 2, 20, 3, 21funcringcsetcALTV2lem2 46647 . . . . . . . . 9 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
60593ad2antr2 1189 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) ∈ 𝑈)
6160adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹𝑌) ∈ 𝑈)
621, 19, 2, 20, 3, 21funcringcsetcALTV2lem2 46647 . . . . . . . . 9 ((𝜑𝑍𝐵) → (𝐹𝑍) ∈ 𝑈)
63623ad2antr3 1190 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) ∈ 𝑈)
6463adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐹𝑍) ∈ 𝑈)
651, 19, 2, 20, 3, 21funcringcsetcALTV2lem1 46646 . . . . . . . . . . . 12 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
66653ad2antr1 1188 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑋) = (Base‘𝑋))
671, 19, 2, 20, 3, 21funcringcsetcALTV2lem1 46646 . . . . . . . . . . . 12 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
68673ad2antr2 1189 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑌) = (Base‘𝑌))
6966, 68feq23d 6700 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7069adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐻:(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(Base‘𝑋)⟶(Base‘𝑌)))
7149, 70mpbird 256 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻:(𝐹𝑋)⟶(𝐹𝑌))
72 simpll 765 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝜑)
73 3simpa 1148 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋𝐵𝑌𝐵))
7473ad2antlr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑋𝐵𝑌𝐵))
75 simprl 769 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐻 ∈ (𝑋 RingHom 𝑌))
761, 19, 2, 20, 3, 21, 22funcringcsetcALTV2lem6 46651 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7772, 74, 75, 76syl3anc 1371 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7877feq1d 6690 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌) ↔ 𝐻:(𝐹𝑋)⟶(𝐹𝑌)))
7971, 78mpbird 256 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑋𝐺𝑌)‘𝐻):(𝐹𝑋)⟶(𝐹𝑌))
801, 19, 2, 20, 3, 21funcringcsetcALTV2lem1 46646 . . . . . . . . . . . 12 ((𝜑𝑍𝐵) → (𝐹𝑍) = (Base‘𝑍))
81803ad2antr3 1190 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐹𝑍) = (Base‘𝑍))
8268, 81feq23d 6700 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8382adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝐾:(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(Base‘𝑌)⟶(Base‘𝑍)))
8452, 83mpbird 256 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾:(𝐹𝑌)⟶(𝐹𝑍))
85 3simpc 1150 . . . . . . . . . . 11 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌𝐵𝑍𝐵))
8685ad2antlr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (𝑌𝐵𝑍𝐵))
87 simprr 771 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → 𝐾 ∈ (𝑌 RingHom 𝑍))
881, 19, 2, 20, 3, 21, 22funcringcsetcALTV2lem6 46651 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐵𝑍𝐵) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍)) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
8972, 86, 87, 88syl3anc 1371 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9089feq1d 6690 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍) ↔ 𝐾:(𝐹𝑌)⟶(𝐹𝑍)))
9184, 90mpbird 256 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → ((𝑌𝐺𝑍)‘𝐾):(𝐹𝑌)⟶(𝐹𝑍))
9219, 26, 55, 58, 61, 64, 79, 91setcco 18017 . . . . . 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 5857 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9492, 93eqtrd 2772 . . . . 5 (((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝐻 ∈ (𝑋 RingHom 𝑌) ∧ 𝐾 ∈ (𝑌 RingHom 𝑍))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
9518, 54, 943eqtr4d 2782 . . . 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 413 . . 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 239 . 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 1117 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cin 3944  cop 4629  cmpt 5225   I cid 5567  cres 5672  ccom 5674  wf 6529  cfv 6533  (class class class)co 7394  cmpo 7396  WUnicwun 10679  Basecbs 17128  Hom chom 17192  compcco 17193  SetCatcsetc 18009  Ringcrg 20016   RingHom crh 20200  RingCatcringc 46613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-er 8688  df-map 8807  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-wun 10681  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-fz 13469  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-hom 17205  df-cco 17206  df-0g 17371  df-resc 17742  df-setc 18010  df-estrc 18058  df-mgm 18545  df-sgrp 18594  df-mnd 18605  df-mhm 18649  df-grp 18799  df-ghm 19058  df-mgp 19949  df-ur 19966  df-ring 20018  df-rnghom 20203  df-ringc 46615
This theorem is referenced by:  funcringcsetcALTV2  46655
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