Theorem coof 7677

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 7056	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
3		coof.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffvelcdmda 7056	. . . 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 3180	. . . . 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 7410	. . . . . 6 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐻‘(𝑏𝑅𝑐)) = (𝐻‘((𝐹‘𝑥)𝑅𝑐)))
9		fveq2 6858	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐻‘𝑏) = (𝐻‘(𝐹‘𝑥)))
10	9	oveq1d 7402	. . . . . 6 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐻‘𝑏)𝑆(𝐻‘𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐)))
11	8, 10	eqeq12d 2745	. . . . 5 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)) ↔ (𝐻‘((𝐹‘𝑥)𝑅𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐))))
12		oveq2 7395	. . . . . . 7 ⊢ (𝑐 = (𝐺‘𝑥) → ((𝐹‘𝑥)𝑅𝑐) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
13	12	fveq2d 6862	. . . . . 6 ⊢ (𝑐 = (𝐺‘𝑥) → (𝐻‘((𝐹‘𝑥)𝑅𝑐)) = (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
14		fveq2 6858	. . . . . . 7 ⊢ (𝑐 = (𝐺‘𝑥) → (𝐻‘𝑐) = (𝐻‘(𝐺‘𝑥)))
15	14	oveq2d 7403	. . . . . 6 ⊢ (𝑐 = (𝐺‘𝑥) → ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘𝑐)) = ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥))))
16	13, 15	eqeq12d 2745	. . . . 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 3600	. . . 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 5200	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥)))))
20	1	ffnd 6689	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
21	3	ffnd 6689	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐴)
22		coof.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23		inidm 4190	. . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴
24		eqidd 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
25		eqidd 2730	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
26	20, 21, 22, 22, 23, 24, 25	offval 7662	. . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
27	26	coeq2d 5826	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
28		coof.h	. . . . 5 ⊢ (𝜑 → 𝐻 Fn 𝐵)
29		dffn3 6700	. . . . 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 7581	. . . . 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 7104	. . 3 ⊢ (𝜑 → (𝐻 ∘ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) = (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
36	27, 35	eqtrd 2764	. 2 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = (𝑥 ∈ 𝐴 ↦ (𝐻‘((𝐹‘𝑥)𝑅(𝐺‘𝑥)))))
37		fnfco 6725	. . . 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 6725	. . . 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 6958	. . . 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 6958	. . . 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 7662	. 2 ⊢ (𝜑 → ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)) = (𝑥 ∈ 𝐴 ↦ ((𝐻‘(𝐹‘𝑥))𝑆(𝐻‘(𝐺‘𝑥)))))
46	19, 36, 45	3eqtr4d 2774	1 ⊢ (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ↦ cmpt 5188  ran crn 5639   ∘ ccom 5642   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653

This theorem is referenced by:  rhmply1vsca  22275