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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcringcsetclem9ALTV Structured version   Visualization version   GIF version
Theorem funcringcsetclem9ALTV 47306
Description: Lemma 9 for funcringcsetcALTV 47307. (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 480 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
5 eqid 2727 . . . . . 6 (Hom β€˜π‘…) = (Hom β€˜π‘…)
6 simpr1 1192 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ 𝑋 ∈ 𝐡)
7 simpr2 1193 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ π‘Œ ∈ 𝐡)
81, 2, 4, 5, 6, 7ringchomALTV 47287 . . . . 5 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋(Hom β€˜π‘…)π‘Œ) = (𝑋 RingHom π‘Œ))
98eleq2d 2814 . . . 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 1194 . . . . . 6 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ 𝑍 ∈ 𝐡)
111, 2, 4, 5, 7, 10ringchomALTV 47287 . . . . 5 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (π‘Œ(Hom β€˜π‘…)𝑍) = (π‘Œ RingHom 𝑍))
1211eleq2d 2814 . . . 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 630 . . 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 20429 . . . . . . . 8 ((𝐾 ∈ (π‘Œ RingHom 𝑍) ∧ 𝐻 ∈ (𝑋 RingHom π‘Œ)) β†’ (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
1514ancoms 458 . . . . . . 7 ((𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍)) β†’ (𝐾 ∘ 𝐻) ∈ (𝑋 RingHom 𝑍))
1615adantl 481 . . . . . 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 7176 . . . . . 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 47302 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋𝐺𝑍) = ( I β†Ύ (𝑋 RingHom 𝑍)))
24233adantr2 1168 . . . . . . 7 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋𝐺𝑍) = ( I β†Ύ (𝑋 RingHom 𝑍)))
2524adantr 480 . . . . . 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 480 . . . . . . 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 2727 . . . . . . 7 (compβ€˜π‘…) = (compβ€˜π‘…)
286adantr 480 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) ∧ (𝐻 ∈ (𝑋 RingHom π‘Œ) ∧ 𝐾 ∈ (π‘Œ RingHom 𝑍))) β†’ 𝑋 ∈ 𝐡)
297adantr 480 . . . . . . 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 480 . . . . . . 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 770 . . . . . . 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 772 . . . . . . 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 47290 . . . . . 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 6898 . . . . 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 2727 . . . . . . 7 (compβ€˜π‘†) = (compβ€˜π‘†)
361, 19, 2, 20, 3, 21funcringcsetclem2ALTV 47299 . . . . . . . . 9 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
37363ad2antr1 1186 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
3837adantr 480 . . . . . . 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 47299 . . . . . . . . 9 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
40393ad2antr2 1187 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
4140adantr 480 . . . . . . 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 47299 . . . . . . . . 9 ((πœ‘ ∧ 𝑍 ∈ 𝐡) β†’ (πΉβ€˜π‘) ∈ π‘ˆ)
43423ad2antr3 1188 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘) ∈ π‘ˆ)
4443adantr 480 . . . . . . 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 2727 . . . . . . . . . . 11 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
46 eqid 2727 . . . . . . . . . . 11 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
4745, 46rhmf 20413 . . . . . . . . . 10 (𝐻 ∈ (𝑋 RingHom π‘Œ) β†’ 𝐻:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
4847ad2antrl 727 . . . . . . . . 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 47298 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
50493ad2antr1 1186 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
511, 19, 2, 20, 3, 21funcringcsetclem1ALTV 47298 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
52513ad2antr2 1187 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
5350, 52feq23d 6711 . . . . . . . . . 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 480 . . . . . . . . 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 257 . . . . . . . 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 766 . . . . . . . . . 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 1146 . . . . . . . . . . 11 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) β†’ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡))
5857ad2antlr 726 . . . . . . . . . 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 47303 . . . . . . . . . 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 1369 . . . . . . . . 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 6701 . . . . . . . 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 257 . . . . . . 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 2727 . . . . . . . . . . 11 (Baseβ€˜π‘) = (Baseβ€˜π‘)
6446, 63rhmf 20413 . . . . . . . . . 10 (𝐾 ∈ (π‘Œ RingHom 𝑍) β†’ 𝐾:(Baseβ€˜π‘Œ)⟢(Baseβ€˜π‘))
6564ad2antll 728 . . . . . . . . 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 47298 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑍 ∈ 𝐡) β†’ (πΉβ€˜π‘) = (Baseβ€˜π‘))
67663ad2antr3 1188 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (πΉβ€˜π‘) = (Baseβ€˜π‘))
6852, 67feq23d 6711 . . . . . . . . . 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 480 . . . . . . . . 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 257 . . . . . . . 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 1148 . . . . . . . . . . 11 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) β†’ (π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡))
7271ad2antlr 726 . . . . . . . . . 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 47303 . . . . . . . . . 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 1369 . . . . . . . . 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 6701 . . . . . . . 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 257 . . . . . . 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 18063 . . . . . 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 5861 . . . . . 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 2767 . . . . 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 2777 . . . 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 412 . . 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 1115 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βŸ¨cop 4630   ↦ cmpt 5225   I cid 5569   β†Ύ cres 5674   ∘ ccom 5676  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  WUnicwun 10715  Basecbs 17171  Hom chom 17235  compcco 17236  SetCatcsetc 18055   RingHom crh 20397  RingCatALTVcringcALTV 47272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-wun 10717  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-plusg 17237  df-hom 17248  df-cco 17249  df-0g 17414  df-setc 18056  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-mhm 18731  df-grp 18884  df-ghm 19159  df-mgp 20066  df-ur 20113  df-ring 20166  df-rhm 20400  df-ringcALTV 47273
This theorem is referenced by:  funcringcsetcALTV  47307
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