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Theorem funcringcsetclem9ALTV 46918
Description: Lemma 9 for funcringcsetcALTV 46919. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTVβ€˜π‘ˆ)
funcringcsetcALTV.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcringcsetcALTV.b 𝐡 = (Baseβ€˜π‘…)
funcringcsetcALTV.c 𝐢 = (Baseβ€˜π‘†)
funcringcsetcALTV.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcringcsetcALTV.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcringcsetcALTV.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem9ALTV ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (𝐻 ∈ (𝑋(Hom β€˜π‘…)π‘Œ) ∧ 𝐾 ∈ (π‘Œ(Hom β€˜π‘…)𝑍))) β†’ ((𝑋𝐺𝑍)β€˜(𝐾(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘…)𝑍)𝐻)) = (((π‘ŒπΊπ‘)β€˜πΎ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩(compβ€˜π‘†)(πΉβ€˜π‘))((π‘‹πΊπ‘Œ)β€˜π»)))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑋   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯   𝑦,𝑋   π‘₯,π‘Œ,𝑦   πœ‘,𝑦   π‘₯,𝑍,𝑦
Allowed substitution hints:   𝐢(𝑦)   𝑅(π‘₯,𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)   𝐻(π‘₯,𝑦)   𝐾(π‘₯,𝑦)
Proof of Theorem funcringcsetclem9ALTV
StepHypRef Expression
1 funcringcsetcALTV.r . . . . . 6 𝑅 = (RingCatALTVβ€˜π‘ˆ)
2 funcringcsetcALTV.b . . . . . 6 𝐡 = (Baseβ€˜π‘…)
3 funcringcsetcALTV.u . . . . . . 7 (πœ‘ β†’ π‘ˆ ∈ WUni)
43adantr 481 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
5 eqid 2732 . . . . . 6 (Hom β€˜π‘…) = (Hom β€˜π‘…)
6 simpr1 1194 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ 𝑋 ∈ 𝐡)
7 simpr2 1195 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ π‘Œ ∈ 𝐡)
81, 2, 4, 5, 6, 7ringchomALTV 46899 . . . . 5 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋(Hom β€˜π‘…)π‘Œ) = (𝑋 RingHom π‘Œ))
98eleq2d 2819 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝐻 ∈ (𝑋(Hom β€˜π‘…)π‘Œ) ↔ 𝐻 ∈ (𝑋 RingHom π‘Œ)))
10 simpr3 1196 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ 𝑍 ∈ 𝐡)
111, 2, 4, 5, 7, 10ringchomALTV 46899 . . . . 5 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (π‘Œ(Hom β€˜π‘…)𝑍) = (π‘Œ RingHom 𝑍))
1211eleq2d 2819 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝐾 ∈ (π‘Œ(Hom β€˜π‘…)𝑍) ↔ 𝐾 ∈ (π‘Œ RingHom 𝑍)))
139, 12anbi12d 631 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝐻 ∈ (𝑋(Hom β€˜π‘…)π‘Œ) ∧ 𝐾 ∈ (π‘Œ(Hom β€˜π‘…)𝑍)) ↔ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))))
14 rhmco 20268 . . . . . . . 8 ((𝐾 ∈ (π‘Œ RingHom 𝑍) ∧ 𝐻 ∈ (𝑋 RingHom π‘Œ)) β†’ (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
1514ancoms 459 . . . . . . 7 ((𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍)) β†’ (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
1615adantl 482 . . . . . 6 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
17 fvresi 7167 . . . . . 6 ((𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍) β†’ (( I β†Ύ (𝑋 RingHom 𝑍))β€˜(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
1816, 17syl 17 . . . . 5 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (( I β†Ύ (𝑋 RingHom 𝑍))β€˜(𝐾 ∘ 𝐻)) = (𝐾 ∘ 𝐻))
19 funcringcsetcALTV.s . . . . . . . . 9 𝑆 = (SetCatβ€˜π‘ˆ)
20 funcringcsetcALTV.c . . . . . . . . 9 𝐢 = (Baseβ€˜π‘†)
21 funcringcsetcALTV.f . . . . . . . . 9 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
22 funcringcsetcALTV.g . . . . . . . . 9 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
231, 19, 2, 20, 3, 21, 22funcringcsetclem5ALTV 46914 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋𝐺𝑍) = ( I β†Ύ (𝑋 RingHom 𝑍)))
24233adantr2 1170 . . . . . . 7 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋𝐺𝑍) = ( I β†Ύ (𝑋 RingHom 𝑍)))
2524adantr 481 . . . . . 6 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (𝑋𝐺𝑍) = ( I β†Ύ (𝑋 RingHom 𝑍)))
264adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ π‘ˆ ∈ WUni)
27 eqid 2732 . . . . . . 7 (compβ€˜π‘…) = (compβ€˜π‘…)
286adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝑋 ∈ 𝐡)
297adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ π‘Œ ∈ 𝐡)
3010adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝑍 ∈ 𝐡)
31 simprl 769 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝐻 ∈ (𝑋 RingHom π‘Œ))
32 simprr 771 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝐾 ∈ (π‘Œ RingHom 𝑍))
331, 2, 26, 27, 28, 29, 30, 31, 32ringccoALTV 46902 . . . . . 6 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (𝐾(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘…)𝑍)𝐻) = (𝐾 ∘ 𝐻))
3425, 33fveq12d 6895 . . . . 5 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ ((𝑋𝐺𝑍)β€˜(𝐾(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘…)𝑍)𝐻)) = (( I β†Ύ (𝑋 RingHom 𝑍))β€˜(𝐾 ∘ 𝐻)))
35 eqid 2732 . . . . . . 7 (compβ€˜π‘†) = (compβ€˜π‘†)
361, 19, 2, 20, 3, 21funcringcsetclem2ALTV 46911 . . . . . . . . 9 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
37363ad2antr1 1188 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
3837adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
391, 19, 2, 20, 3, 21funcringcsetclem2ALTV 46911 . . . . . . . . 9 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
40393ad2antr2 1189 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
4140adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
421, 19, 2, 20, 3, 21funcringcsetclem2ALTV 46911 . . . . . . . . 9 ((πœ‘ ∧ 𝑍 ∈ 𝐡) β†’ (πΉβ€˜π‘) ∈ π‘ˆ)
43423ad2antr3 1190 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘) ∈ π‘ˆ)
4443adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (πΉβ€˜π‘) ∈ π‘ˆ)
45 eqid 2732 . . . . . . . . . . 11 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
46 eqid 2732 . . . . . . . . . . 11 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
4745, 46rhmf 20255 . . . . . . . . . 10 (𝐻 ∈ (𝑋 RingHom π‘Œ) β†’ 𝐻:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
4847ad2antrl 726 . . . . . . . . 9 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝐻:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
491, 19, 2, 20, 3, 21funcringcsetclem1ALTV 46910 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
50493ad2antr1 1188 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
511, 19, 2, 20, 3, 21funcringcsetclem1ALTV 46910 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
52513ad2antr2 1189 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
5350, 52feq23d 6709 . . . . . . . . . 10 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝐻:(πΉβ€˜π‘‹)⟢(πΉβ€˜π‘Œ) ↔ 𝐻:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)))
5453adantr 481 . . . . . . . . 9 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (𝐻:(πΉβ€˜π‘‹)⟢(πΉβ€˜π‘Œ) ↔ 𝐻:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)))
5548, 54mpbird 256 . . . . . . . 8 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝐻:(πΉβ€˜π‘‹)⟢(πΉβ€˜π‘Œ))
56 simpll 765 . . . . . . . . . 10 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ πœ‘)
57 3simpa 1148 . . . . . . . . . . 11 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) β†’ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
5857ad2antlr 725 . . . . . . . . . 10 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
591, 19, 2, 20, 3, 21, 22funcringcsetclem6ALTV 46915 . . . . . . . . . 10 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝐻 ∈ (𝑋 RingHom π‘Œ)) β†’ ((π‘‹πΊπ‘Œ)β€˜π») = 𝐻)
6056, 58, 31, 59syl3anc 1371 . . . . . . . . 9 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ ((π‘‹πΊπ‘Œ)β€˜π») = 𝐻)
6160feq1d 6699 . . . . . . . 8 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (((π‘‹πΊπ‘Œ)β€˜π»):(πΉβ€˜π‘‹)⟢(πΉβ€˜π‘Œ) ↔ 𝐻:(πΉβ€˜π‘‹)⟢(πΉβ€˜π‘Œ)))
6255, 61mpbird 256 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ ((π‘‹πΊπ‘Œ)β€˜π»):(πΉβ€˜π‘‹)⟢(πΉβ€˜π‘Œ))
63 eqid 2732 . . . . . . . . . . 11 (Baseβ€˜π‘) = (Baseβ€˜π‘)
6446, 63rhmf 20255 . . . . . . . . . 10 (𝐾 ∈ (π‘Œ RingHom 𝑍) β†’ 𝐾:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘))
6564ad2antll 727 . . . . . . . . 9 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝐾:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘))
661, 19, 2, 20, 3, 21funcringcsetclem1ALTV 46910 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑍 ∈ 𝐡) β†’ (πΉβ€˜π‘) = (Baseβ€˜π‘))
67663ad2antr3 1190 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘) = (Baseβ€˜π‘))
6852, 67feq23d 6709 . . . . . . . . . 10 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝐾:(πΉβ€˜π‘Œ)⟢(πΉβ€˜π‘) ↔ 𝐾:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘)))
6968adantr 481 . . . . . . . . 9 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (𝐾:(πΉβ€˜π‘Œ)⟢(πΉβ€˜π‘) ↔ 𝐾:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘)))
7065, 69mpbird 256 . . . . . . . 8 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝐾:(πΉβ€˜π‘Œ)⟢(πΉβ€˜π‘))
71 3simpc 1150 . . . . . . . . . . 11 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) β†’ (π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡))
7271ad2antlr 725 . . . . . . . . . 10 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡))
731, 19, 2, 20, 3, 21, 22funcringcsetclem6ALTV 46915 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍)) β†’ ((π‘ŒπΊπ‘)β€˜πΎ) = 𝐾)
7456, 72, 32, 73syl3anc 1371 . . . . . . . . 9 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ ((π‘ŒπΊπ‘)β€˜πΎ) = 𝐾)
7574feq1d 6699 . . . . . . . 8 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (((π‘ŒπΊπ‘)β€˜πΎ):(πΉβ€˜π‘Œ)⟢(πΉβ€˜π‘) ↔ 𝐾:(πΉβ€˜π‘Œ)⟢(πΉβ€˜π‘)))
7670, 75mpbird 256 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ ((π‘ŒπΊπ‘)β€˜πΎ):(πΉβ€˜π‘Œ)⟢(πΉβ€˜π‘))
7719, 26, 35, 38, 41, 44, 62, 76setcco 18029 . . . . . 6 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (((π‘ŒπΊπ‘)β€˜πΎ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩(compβ€˜π‘†)(πΉβ€˜π‘))((π‘‹πΊπ‘Œ)β€˜π»)) = (((π‘ŒπΊπ‘)β€˜πΎ) ∘ ((π‘‹πΊπ‘Œ)β€˜π»)))
7874, 60coeq12d 5862 . . . . . 6 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (((π‘ŒπΊπ‘)β€˜πΎ) ∘ ((π‘‹πΊπ‘Œ)β€˜π»)) = (𝐾 ∘ 𝐻))
7977, 78eqtrd 2772 . . . . 5 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ (((π‘ŒπΊπ‘)β€˜πΎ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩(compβ€˜π‘†)(πΉβ€˜π‘))((π‘‹πΊπ‘Œ)β€˜π»)) = (𝐾 ∘ 𝐻))
8018, 34, 793eqtr4d 2782 . . . 4 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ ((𝑋𝐺𝑍)β€˜(𝐾(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘…)𝑍)𝐻)) = (((π‘ŒπΊπ‘)β€˜πΎ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩(compβ€˜π‘†)(πΉβ€˜π‘))((π‘‹πΊπ‘Œ)β€˜π»)))
8180ex 413 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍)) β†’ ((𝑋𝐺𝑍)β€˜(𝐾(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘…)𝑍)𝐻)) = (((π‘ŒπΊπ‘)β€˜πΎ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩(compβ€˜π‘†)(πΉβ€˜π‘))((π‘‹πΊπ‘Œ)β€˜π»))))
8213, 81sylbid 239 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝐻 ∈ (𝑋(Hom β€˜π‘…)π‘Œ) ∧ 𝐾 ∈ (π‘Œ(Hom β€˜π‘…)𝑍)) β†’ ((𝑋𝐺𝑍)β€˜(𝐾(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘…)𝑍)𝐻)) = (((π‘ŒπΊπ‘)β€˜πΎ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩(compβ€˜π‘†)(πΉβ€˜π‘))((π‘‹πΊπ‘Œ)β€˜π»))))
83823impia 1117 1 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (𝐻 ∈ (𝑋(Hom β€˜π‘…)π‘Œ) ∧ 𝐾 ∈ (π‘Œ(Hom β€˜π‘…)𝑍))) β†’ ((𝑋𝐺𝑍)β€˜(𝐾(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜π‘…)𝑍)𝐻)) = (((π‘ŒπΊπ‘)β€˜πΎ)(⟨(πΉβ€˜π‘‹), (πΉβ€˜π‘Œ)⟩(compβ€˜π‘†)(πΉβ€˜π‘))((π‘‹πΊπ‘Œ)β€˜π»)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βŸ¨cop 4633   ↦ cmpt 5230   I cid 5572   β†Ύ cres 5677   ∘ ccom 5679  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  WUnicwun 10691  Basecbs 17140  Hom chom 17204  compcco 17205  SetCatcsetc 18021   RingHom crh 20240  RingCatALTVcringcALTV 46855
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-wun 10693  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-hom 17217  df-cco 17218  df-0g 17383  df-setc 18022  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-grp 18818  df-ghm 19084  df-mgp 19982  df-ur 19999  df-ring 20051  df-rnghom 20243  df-ringcALTV 46857
This theorem is referenced by:  funcringcsetcALTV  46919
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