Theorem coof 7640

Description: The composition of a homomorphism with a function operation. (Contributed by SN, 20-May-2025.)

Hypotheses

Ref	Expression
coof.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
coof.g	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
coof.h	⊢ (𝜑 → 𝐻 Fn 𝐵)
coof.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
coof.1	⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵)
coof.2	⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))

Assertion

Ref	Expression
coof	⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)))

Distinct variable groups:   𝐵,𝑏,𝑐   𝐹,𝑏,𝑐   𝐺,𝑏,𝑐   𝐻,𝑏,𝑐   𝑅,𝑏,𝑐   𝑆,𝑏,𝑐   𝜑,𝑏,𝑐

Allowed substitution hints:   𝐴(𝑏,𝑐)   𝑉(𝑏,𝑐)

Proof of Theorem coof

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coof.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffvelcdmda 7023	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
3		coof.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffvelcdmda 7023	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
5		coof.2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))
6	5	ralrimivva 3176	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))
8		fvoveq1 7375	. . . . . 6 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐻‘(𝑏𝑅𝑐)) = (𝐻‘((𝐹‘𝑥)𝑅𝑐)))
9		fveq2 6828	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐻‘𝑏) = (𝐻‘(𝐹‘𝑥)))
10	9	oveq1d 7367	. . . . . 6 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐻‘𝑏)𝑆(𝐻‘𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐)))
11	8, 10	eqeq12d 2749	. . . . 5 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)) ↔ (𝐻‘((𝐹‘𝑥)𝑅𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐))))
12		oveq2 7360	. . . . . . 7 ⊢ (𝑐 = (𝐺‘𝑥) → ((𝐹‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
13	12	fveq2d 6832	. . . . . 6 ⊢ (𝑐 = (𝐺‘𝑥) → (𝐻‘((𝐹‘𝑥)𝑅𝑐)) = (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
14		fveq2 6828	. . . . . . 7 ⊢ (𝑐 = (𝐺‘𝑥) → (𝐻‘𝑐) = (𝐻‘(𝐺‘𝑥)))
15	14	oveq2d 7368	. . . . . 6 ⊢ (𝑐 = (𝐺‘𝑥) → ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥))))
16	13, 15	eqeq12d 2749	. . . . 5 ⊢ (𝑐 = (𝐺‘𝑥) → ((𝐻‘((𝐹‘𝑥)𝑅𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐)) ↔ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥)))))
17	11, 16	rspc2va 3585	. . . 4 ⊢ ((((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐵) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐))) → (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥))))
18	2, 4, 7, 17	syl21anc 837	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥))))
19	18	mpteq2dva 5186	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥)))))
20	1	ffnd 6657	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
21	3	ffnd 6657	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
22		coof.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23		inidm 4176	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
24		eqidd 2734	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
25		eqidd 2734	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
26	20, 21, 22, 22, 23, 24, 25	offval 7625	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
27	26	coeq2d 5806	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
28		coof.h	. . . . 5 ⊢ (𝜑 → 𝐻 Fn 𝐵)
29		dffn3 6668	. . . . 5 ⊢ (𝐻 Fn 𝐵 ↔ 𝐻:𝐵⟶ran 𝐻)
30	28, 29	sylib 218	. . . 4 ⊢ (𝜑 → 𝐻:𝐵⟶ran 𝐻)
31	2, 4	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐵))
32		coof.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵)
33	32	caovclg 7544	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐵)) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ 𝐵)
34	31, 33	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ 𝐵)
35	30, 34	cofmpt 7071	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
36	27, 35	eqtrd 2768	. 2 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
37		fnfco 6693	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
38	28, 1, 37	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) Fn 𝐴)
39		fnfco 6693	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐺:𝐴⟶𝐵) → (𝐻 ∘ 𝐺) Fn 𝐴)
40	28, 3, 39	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐺) Fn 𝐴)
41		fvco2 6925	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑥) = (𝐻‘(𝐹‘𝑥)))
42	20, 41	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑥) = (𝐻‘(𝐹‘𝑥)))
43		fvco2 6925	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐺)‘𝑥) = (𝐻‘(𝐺‘𝑥)))
44	21, 43	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐺)‘𝑥) = (𝐻‘(𝐺‘𝑥)))
45	38, 40, 22, 22, 23, 42, 44	offval 7625	. 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)) = (𝑥 ∈ 𝐴 ↦ ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥)))))
46	19, 36, 45	3eqtr4d 2778	1 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ↦ cmpt 5174  ran crn 5620   ∘ ccom 5623   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ∘_f cof 7614

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-pr 5372

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-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-of 7616

This theorem is referenced by:  rhmply1vsca  22304