Theorem coof 7718

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 7085	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
3		coof.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffvelcdmda 7085	. . . 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 3205	. . . . 5 ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))
8		fvoveq1 7444	. . . . . 6 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐻‘(𝑏𝑅𝑐)) = (𝐻‘((𝐹‘𝑥)𝑅𝑐)))
9		fveq2 6886	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐻‘𝑏) = (𝐻‘(𝐹‘𝑥)))
10	9	oveq1d 7436	. . . . . 6 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐻‘𝑏)𝑆(𝐻‘𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐)))
11	8, 10	eqeq12d 2776	. . . . 5 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)) ↔ (𝐻‘((𝐹‘𝑥)𝑅𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐))))
12		oveq2 7429	. . . . . . 7 ⊢ (𝑐 = (𝐺‘𝑥) → ((𝐹‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
13	12	fveq2d 6890	. . . . . 6 ⊢ (𝑐 = (𝐺‘𝑥) → (𝐻‘((𝐹‘𝑥)𝑅𝑐)) = (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
14		fveq2 6886	. . . . . . 7 ⊢ (𝑐 = (𝐺‘𝑥) → (𝐻‘𝑐) = (𝐻‘(𝐺‘𝑥)))
15	14	oveq2d 7437	. . . . . 6 ⊢ (𝑐 = (𝐺‘𝑥) → ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥))))
16	13, 15	eqeq12d 2776	. . . . 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 3587	. . . 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 851	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥))))
19	18	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥)))))
20	1	ffnd 6711	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
21	3	ffnd 6711	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
22		coof.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23		inidm 4123	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
24		eqidd 2761	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
25		eqidd 2761	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
26	20, 21, 22, 22, 23, 24, 25	offval 7703	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
27	26	coeq2d 5839	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
28		coof.h	. . . . 5 ⊢ (𝜑 → 𝐻 Fn 𝐵)
29		dffn3 6723	. . . . 5 ⊢ (𝐻 Fn 𝐵 ↔ 𝐻:𝐵⟶ran 𝐻)
30	28, 29	sylib 221	. . . 4 ⊢ (𝜑 → 𝐻:𝐵⟶ran 𝐻)
31	2, 4	jca 521	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐵))
32		coof.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵)
33	32	caovclg 7614	. . . . 5 ⊢ ((𝜑 ∧ ((𝐹‘𝑥) ∈ 𝐵 ∧ (𝐺‘𝑥) ∈ 𝐵)) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ 𝐵)
34	31, 33	syldan 603	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ 𝐵)
35	30, 34	cofmpt 7134	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
36	27, 35	eqtrd 2795	. 2 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
37		fnfco 6748	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐻 ∘ 𝐹) Fn 𝐴)
38	28, 1, 37	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐹) Fn 𝐴)
39		fnfco 6748	. . . 4 ⊢ ((𝐻 Fn 𝐵 ∧ 𝐺:𝐴⟶𝐵) → (𝐻 ∘ 𝐺) Fn 𝐴)
40	28, 3, 39	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐺) Fn 𝐴)
41		fvco2 6983	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑥) = (𝐻‘(𝐹‘𝑥)))
42	20, 41	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐹)‘𝑥) = (𝐻‘(𝐹‘𝑥)))
43		fvco2 6983	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐺)‘𝑥) = (𝐻‘(𝐺‘𝑥)))
44	21, 43	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐻 ∘ 𝐺)‘𝑥) = (𝐻‘(𝐺‘𝑥)))
45	38, 40, 22, 22, 23, 42, 44	offval 7703	. 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)) = (𝑥 ∈ 𝐴 ↦ ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥)))))
46	19, 36, 45	3eqtr4d 2805	1 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ↦ cmpt 5185  ran crn 5651   ∘ ccom 5654   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ∘_f cof 7692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-of 7694

This theorem is used by:  rhmply1vsca  22709