MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcsetcestrclem9 Structured version   Visualization version   GIF version

Theorem funcsetcestrclem9 17403
Description: Lemma 9 for funcsetcestrc 17404. (Contributed by AV, 28-Mar-2020.)
Hypotheses
Ref Expression
funcsetcestrc.s 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c 𝐶 = (Base‘𝑆)
funcsetcestrc.f (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u (𝜑𝑈 ∈ WUni)
funcsetcestrc.o (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
funcsetcestrc.e 𝐸 = (ExtStrCat‘𝑈)
Assertion
Ref Expression
funcsetcestrclem9 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑋   𝜑,𝑥   𝑦,𝐶,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦   𝑥,𝑍,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem funcsetcestrclem9
StepHypRef Expression
1 funcsetcestrc.s . . . . . 6 𝑆 = (SetCat‘𝑈)
2 funcsetcestrc.u . . . . . . 7 (𝜑𝑈 ∈ WUni)
32adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑈 ∈ WUni)
4 eqid 2826 . . . . . 6 (Hom ‘𝑆) = (Hom ‘𝑆)
51, 2setcbas 17328 . . . . . . . . . . 11 (𝜑𝑈 = (Base‘𝑆))
6 funcsetcestrc.c . . . . . . . . . . 11 𝐶 = (Base‘𝑆)
75, 6syl6reqr 2880 . . . . . . . . . 10 (𝜑𝐶 = 𝑈)
87eleq2d 2903 . . . . . . . . 9 (𝜑 → (𝑋𝐶𝑋𝑈))
98biimpcd 250 . . . . . . . 8 (𝑋𝐶 → (𝜑𝑋𝑈))
1093ad2ant1 1127 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑋𝑈))
1110impcom 408 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑋𝑈)
127eleq2d 2903 . . . . . . . . 9 (𝜑 → (𝑌𝐶𝑌𝑈))
1312biimpcd 250 . . . . . . . 8 (𝑌𝐶 → (𝜑𝑌𝑈))
14133ad2ant2 1128 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑌𝑈))
1514impcom 408 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑌𝑈)
161, 3, 4, 11, 15setchom 17330 . . . . 5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑋(Hom ‘𝑆)𝑌) = (𝑌m 𝑋))
1716eleq2d 2903 . . . 4 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ↔ 𝐻 ∈ (𝑌m 𝑋)))
187eleq2d 2903 . . . . . . . . 9 (𝜑 → (𝑍𝐶𝑍𝑈))
1918biimpcd 250 . . . . . . . 8 (𝑍𝐶 → (𝜑𝑍𝑈))
20193ad2ant3 1129 . . . . . . 7 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝜑𝑍𝑈))
2120impcom 408 . . . . . 6 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → 𝑍𝑈)
221, 3, 4, 15, 21setchom 17330 . . . . 5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑌(Hom ‘𝑆)𝑍) = (𝑍m 𝑌))
2322eleq2d 2903 . . . 4 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍) ↔ 𝐾 ∈ (𝑍m 𝑌)))
2417, 23anbi12d 630 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → ((𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍)) ↔ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))))
25 elmapi 8418 . . . . . . . . 9 (𝐾 ∈ (𝑍m 𝑌) → 𝐾:𝑌𝑍)
26 elmapi 8418 . . . . . . . . 9 (𝐻 ∈ (𝑌m 𝑋) → 𝐻:𝑋𝑌)
27 fco 6528 . . . . . . . . 9 ((𝐾:𝑌𝑍𝐻:𝑋𝑌) → (𝐾𝐻):𝑋𝑍)
2825, 26, 27syl2anr 596 . . . . . . . 8 ((𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌)) → (𝐾𝐻):𝑋𝑍)
2928adantl 482 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐾𝐻):𝑋𝑍)
30 elmapg 8409 . . . . . . . . . 10 ((𝑍𝐶𝑋𝐶) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3130ancoms 459 . . . . . . . . 9 ((𝑋𝐶𝑍𝐶) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
32313adant2 1125 . . . . . . . 8 ((𝑋𝐶𝑌𝐶𝑍𝐶) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3332ad2antlr 723 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝐾𝐻) ∈ (𝑍m 𝑋) ↔ (𝐾𝐻):𝑋𝑍))
3429, 33mpbird 258 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐾𝐻) ∈ (𝑍m 𝑋))
35 fvresi 6931 . . . . . 6 ((𝐾𝐻) ∈ (𝑍m 𝑋) → (( I ↾ (𝑍m 𝑋))‘(𝐾𝐻)) = (𝐾𝐻))
3634, 35syl 17 . . . . 5 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (( I ↾ (𝑍m 𝑋))‘(𝐾𝐻)) = (𝐾𝐻))
37 funcsetcestrc.f . . . . . . . . 9 (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
38 funcsetcestrc.o . . . . . . . . 9 (𝜑 → ω ∈ 𝑈)
39 funcsetcestrc.g . . . . . . . . 9 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦m 𝑥))))
401, 6, 37, 2, 38, 39funcsetcestrclem5 17399 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑍𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍m 𝑋)))
41403adantr2 1164 . . . . . . 7 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝑋𝐺𝑍) = ( I ↾ (𝑍m 𝑋)))
4241adantr 481 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝑋𝐺𝑍) = ( I ↾ (𝑍m 𝑋)))
433adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑈 ∈ WUni)
44 eqid 2826 . . . . . . 7 (comp‘𝑆) = (comp‘𝑆)
4511adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑋𝑈)
4615adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑌𝑈)
4721adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝑍𝑈)
4826ad2antrl 724 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐻:𝑋𝑌)
4925ad2antll 725 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐾:𝑌𝑍)
501, 43, 44, 45, 46, 47, 48, 49setcco 17333 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻) = (𝐾𝐻))
5142, 50fveq12d 6674 . . . . 5 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (( I ↾ (𝑍m 𝑋))‘(𝐾𝐻)))
52 funcsetcestrc.e . . . . . . 7 𝐸 = (ExtStrCat‘𝑈)
53 eqid 2826 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
541, 6, 37, 2, 38funcsetcestrclem2 17395 . . . . . . . . 9 ((𝜑𝑋𝐶) → (𝐹𝑋) ∈ 𝑈)
55543ad2antr1 1182 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑋) ∈ 𝑈)
5655adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐹𝑋) ∈ 𝑈)
571, 6, 37, 2, 38funcsetcestrclem2 17395 . . . . . . . . 9 ((𝜑𝑌𝐶) → (𝐹𝑌) ∈ 𝑈)
58573ad2antr2 1183 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑌) ∈ 𝑈)
5958adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐹𝑌) ∈ 𝑈)
601, 6, 37, 2, 38funcsetcestrclem2 17395 . . . . . . . . 9 ((𝜑𝑍𝐶) → (𝐹𝑍) ∈ 𝑈)
61603ad2antr3 1184 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑍) ∈ 𝑈)
6261adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝐹𝑍) ∈ 𝑈)
63 eqid 2826 . . . . . . 7 (Base‘(𝐹𝑋)) = (Base‘(𝐹𝑋))
64 eqid 2826 . . . . . . 7 (Base‘(𝐹𝑌)) = (Base‘(𝐹𝑌))
65 eqid 2826 . . . . . . 7 (Base‘(𝐹𝑍)) = (Base‘(𝐹𝑍))
66 simpll 763 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝜑)
67 3simpa 1142 . . . . . . . . . . 11 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝑋𝐶𝑌𝐶))
6867ad2antlr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝑋𝐶𝑌𝐶))
69 simprl 767 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐻 ∈ (𝑌m 𝑋))
701, 6, 37, 2, 38, 39funcsetcestrclem6 17400 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶) ∧ 𝐻 ∈ (𝑌m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
7166, 68, 69, 70syl3anc 1365 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
721, 6, 37funcsetcestrclem1 17394 . . . . . . . . . . . . 13 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
73723ad2antr1 1182 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
7473fveq2d 6671 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑋)) = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
75 eqid 2826 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑋⟩} = {⟨(Base‘ndx), 𝑋⟩}
76751strbas 16589 . . . . . . . . . . . . . 14 (𝑋𝐶𝑋 = (Base‘{⟨(Base‘ndx), 𝑋⟩}))
7776eqcomd 2832 . . . . . . . . . . . . 13 (𝑋𝐶 → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
78773ad2ant1 1127 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
7978adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑋⟩}) = 𝑋)
8074, 79eqtrd 2861 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑋)) = 𝑋)
8180adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (Base‘(𝐹𝑋)) = 𝑋)
821, 6, 37funcsetcestrclem1 17394 . . . . . . . . . . . . 13 ((𝜑𝑌𝐶) → (𝐹𝑌) = {⟨(Base‘ndx), 𝑌⟩})
83823ad2antr2 1183 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑌) = {⟨(Base‘ndx), 𝑌⟩})
8483fveq2d 6671 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑌)) = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
85 eqid 2826 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑌⟩} = {⟨(Base‘ndx), 𝑌⟩}
86851strbas 16589 . . . . . . . . . . . . . 14 (𝑌𝐶𝑌 = (Base‘{⟨(Base‘ndx), 𝑌⟩}))
8786eqcomd 2832 . . . . . . . . . . . . 13 (𝑌𝐶 → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
88873ad2ant2 1128 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
8988adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑌⟩}) = 𝑌)
9084, 89eqtrd 2861 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑌)) = 𝑌)
9190adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (Base‘(𝐹𝑌)) = 𝑌)
9271, 81, 91feq123d 6500 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹𝑋))⟶(Base‘(𝐹𝑌)) ↔ 𝐻:𝑋𝑌))
9348, 92mpbird 258 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑌)‘𝐻):(Base‘(𝐹𝑋))⟶(Base‘(𝐹𝑌)))
94 3simpc 1144 . . . . . . . . . . 11 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (𝑌𝐶𝑍𝐶))
9594ad2antlr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (𝑌𝐶𝑍𝐶))
96 simprr 769 . . . . . . . . . 10 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → 𝐾 ∈ (𝑍m 𝑌))
971, 6, 37, 2, 38, 39funcsetcestrclem6 17400 . . . . . . . . . 10 ((𝜑 ∧ (𝑌𝐶𝑍𝐶) ∧ 𝐾 ∈ (𝑍m 𝑌)) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
9866, 95, 96, 97syl3anc 1365 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑌𝐺𝑍)‘𝐾) = 𝐾)
991, 6, 37funcsetcestrclem1 17394 . . . . . . . . . . . . 13 ((𝜑𝑍𝐶) → (𝐹𝑍) = {⟨(Base‘ndx), 𝑍⟩})
100993ad2antr3 1184 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (𝐹𝑍) = {⟨(Base‘ndx), 𝑍⟩})
101100fveq2d 6671 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑍)) = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
102 eqid 2826 . . . . . . . . . . . . . . 15 {⟨(Base‘ndx), 𝑍⟩} = {⟨(Base‘ndx), 𝑍⟩}
1031021strbas 16589 . . . . . . . . . . . . . 14 (𝑍𝐶𝑍 = (Base‘{⟨(Base‘ndx), 𝑍⟩}))
104103eqcomd 2832 . . . . . . . . . . . . 13 (𝑍𝐶 → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
1051043ad2ant3 1129 . . . . . . . . . . . 12 ((𝑋𝐶𝑌𝐶𝑍𝐶) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
106105adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘{⟨(Base‘ndx), 𝑍⟩}) = 𝑍)
107101, 106eqtrd 2861 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → (Base‘(𝐹𝑍)) = 𝑍)
108107adantr 481 . . . . . . . . 9 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (Base‘(𝐹𝑍)) = 𝑍)
10998, 91, 108feq123d 6500 . . . . . . . 8 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾):(Base‘(𝐹𝑌))⟶(Base‘(𝐹𝑍)) ↔ 𝐾:𝑌𝑍))
11049, 109mpbird 258 . . . . . . 7 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑌𝐺𝑍)‘𝐾):(Base‘(𝐹𝑌))⟶(Base‘(𝐹𝑍)))
11152, 43, 53, 56, 59, 62, 63, 64, 65, 93, 110estrcco 17370 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)))
11298, 71coeq12d 5734 . . . . . 6 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾) ∘ ((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
113111, 112eqtrd 2861 . . . . 5 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)) = (𝐾𝐻))
11436, 51, 1133eqtr4d 2871 . . . 4 (((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) ∧ (𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
115114ex 413 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → ((𝐻 ∈ (𝑌m 𝑋) ∧ 𝐾 ∈ (𝑍m 𝑌)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
11624, 115sylbid 241 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶)) → ((𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍)) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻))))
1171163impia 1111 1 ((𝜑 ∧ (𝑋𝐶𝑌𝐶𝑍𝐶) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑆)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑆)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑆)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝐸)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {csn 4564  cop 4570  cmpt 5143   I cid 5458  cres 5556  ccom 5558  wf 6348  cfv 6352  (class class class)co 7148  cmpo 7150  ωcom 7568  m cmap 8396  WUnicwun 10111  ndxcnx 16470  Basecbs 16473  Hom chom 16566  compcco 16567  SetCatcsetc 17325  ExtStrCatcestrc 17362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-inf2 9093  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-omul 8098  df-er 8279  df-ec 8281  df-qs 8285  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-wun 10113  df-ni 10283  df-pli 10284  df-mi 10285  df-lti 10286  df-plpq 10319  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-plq 10325  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329  df-np 10392  df-plp 10394  df-ltp 10396  df-enr 10466  df-nr 10467  df-c 10532  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-hom 16579  df-cco 16580  df-setc 17326  df-estrc 17363
This theorem is referenced by:  funcsetcestrc  17404
  Copyright terms: Public domain W3C validator