Theorem fuco23alem 49476

Description: The naturality property (nati 17867) in category 𝐸. (Contributed by Zhi Wang, 3-Oct-2025.)

Hypotheses

Ref	Expression
fuco23a.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
fuco23a.b	⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
fuco23a.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
fuco23alem.o	⊢ · = (comp‘𝐸)

Assertion

Ref	Expression
fuco23alem	⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩ · (𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩ · (𝑅‘(𝑀‘𝑋)))(𝐵‘(𝐹‘𝑋))))

Proof of Theorem fuco23alem

Step	Hyp	Ref	Expression
1		eqid 2733	. 2 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
2		fuco23a.b	. 2 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
3		eqid 2733	. 2 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		eqid 2733	. 2 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		fuco23alem.o	. 2 ⊢ · = (comp‘𝐸)
6		eqid 2733	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
7		eqid 2733	. . . . 5 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8		fuco23a.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
9	7, 8	natrcl2 49349	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
10	6, 3, 9	funcf1 17775	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝐶)⟶(Base‘𝐷))
11		fuco23a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))
12	10, 11	ffvelcdmd 7024	. 2 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
13	7, 8	natrcl3 49350	. . . 4 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
14	6, 3, 13	funcf1 17775	. . 3 ⊢ (𝜑 → 𝑀:(Base‘𝐶)⟶(Base‘𝐷))
15	14, 11	ffvelcdmd 7024	. 2 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝐷))
16	7, 8, 6, 4, 11	natcl 17865	. 2 ⊢ (𝜑 → (𝐴‘𝑋) ∈ ((𝐹‘𝑋)(Hom ‘𝐷)(𝑀‘𝑋)))
17	1, 2, 3, 4, 5, 12, 15, 16	nati 17867	1 ⊢ (𝜑 → ((𝐵‘(𝑀‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝐾‘(𝑀‘𝑋))⟩ · (𝑅‘(𝑀‘𝑋)))(((𝐹‘𝑋)𝐿(𝑀‘𝑋))‘(𝐴‘𝑋))) = ((((𝐹‘𝑋)𝑆(𝑀‘𝑋))‘(𝐴‘𝑋))(⟨(𝐾‘(𝐹‘𝑋)), (𝑅‘(𝐹‘𝑋))⟩ · (𝑅‘(𝑀‘𝑋)))(𝐵‘(𝐹‘𝑋))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4581  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  Hom chom 17174  compcco 17175   Nat cnat 17853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-ixp 8828  df-func 17767  df-nat 17855

This theorem is referenced by:  fuco23a  49477  fucoco  49482