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Theorem coof 7721
Description: The composition of a homomorphism with a function operation. (Contributed by SN, 20-May-2025.)
Hypotheses
Ref Expression
coof.f (𝜑𝐹:𝐴𝐵)
coof.g (𝜑𝐺:𝐴𝐵)
coof.h (𝜑𝐻 Fn 𝐵)
coof.a (𝜑𝐴𝑉)
coof.1 ((𝜑 ∧ (𝑏𝐵𝑐𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵)
coof.2 ((𝜑 ∧ (𝑏𝐵𝑐𝐵)) → (𝐻‘(𝑏𝑅𝑐)) = ((𝐻𝑏)𝑆(𝐻𝑐)))
Assertion
Ref Expression
coof (𝜑 → (𝐻 ∘ (𝐹f 𝑅𝐺)) = ((𝐻𝐹) ∘f 𝑆(𝐻𝐺)))
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.
StepHypRef Expression
1 coof.f . . . . 5 (𝜑𝐹:𝐴𝐵)
21ffvelcdmda 7104 . . . 4 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
3 coof.g . . . . 5 (𝜑𝐺:𝐴𝐵)
43ffvelcdmda 7104 . . . 4 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
5 coof.2 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝐵)) → (𝐻‘(𝑏𝑅𝑐)) = ((𝐻𝑏)𝑆(𝐻𝑐)))
65ralrimivva 3200 . . . . 5 (𝜑 → ∀𝑏𝐵𝑐𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻𝑏)𝑆(𝐻𝑐)))
76adantr 480 . . . 4 ((𝜑𝑥𝐴) → ∀𝑏𝐵𝑐𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻𝑏)𝑆(𝐻𝑐)))
8 fvoveq1 7454 . . . . . 6 (𝑏 = (𝐹𝑥) → (𝐻‘(𝑏𝑅𝑐)) = (𝐻‘((𝐹𝑥)𝑅𝑐)))
9 fveq2 6907 . . . . . . 7 (𝑏 = (𝐹𝑥) → (𝐻𝑏) = (𝐻‘(𝐹𝑥)))
109oveq1d 7446 . . . . . 6 (𝑏 = (𝐹𝑥) → ((𝐻𝑏)𝑆(𝐻𝑐)) = ((𝐻‘(𝐹𝑥))𝑆(𝐻𝑐)))
118, 10eqeq12d 2751 . . . . 5 (𝑏 = (𝐹𝑥) → ((𝐻‘(𝑏𝑅𝑐)) = ((𝐻𝑏)𝑆(𝐻𝑐)) ↔ (𝐻‘((𝐹𝑥)𝑅𝑐)) = ((𝐻‘(𝐹𝑥))𝑆(𝐻𝑐))))
12 oveq2 7439 . . . . . . 7 (𝑐 = (𝐺𝑥) → ((𝐹𝑥)𝑅𝑐) = ((𝐹𝑥)𝑅(𝐺𝑥)))
1312fveq2d 6911 . . . . . 6 (𝑐 = (𝐺𝑥) → (𝐻‘((𝐹𝑥)𝑅𝑐)) = (𝐻‘((𝐹𝑥)𝑅(𝐺𝑥))))
14 fveq2 6907 . . . . . . 7 (𝑐 = (𝐺𝑥) → (𝐻𝑐) = (𝐻‘(𝐺𝑥)))
1514oveq2d 7447 . . . . . 6 (𝑐 = (𝐺𝑥) → ((𝐻‘(𝐹𝑥))𝑆(𝐻𝑐)) = ((𝐻‘(𝐹𝑥))𝑆(𝐻‘(𝐺𝑥))))
1613, 15eqeq12d 2751 . . . . 5 (𝑐 = (𝐺𝑥) → ((𝐻‘((𝐹𝑥)𝑅𝑐)) = ((𝐻‘(𝐹𝑥))𝑆(𝐻𝑐)) ↔ (𝐻‘((𝐹𝑥)𝑅(𝐺𝑥))) = ((𝐻‘(𝐹𝑥))𝑆(𝐻‘(𝐺𝑥)))))
1711, 16rspc2va 3634 . . . 4 ((((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐵) ∧ ∀𝑏𝐵𝑐𝐵 (𝐻‘(𝑏𝑅𝑐)) = ((𝐻𝑏)𝑆(𝐻𝑐))) → (𝐻‘((𝐹𝑥)𝑅(𝐺𝑥))) = ((𝐻‘(𝐹𝑥))𝑆(𝐻‘(𝐺𝑥))))
182, 4, 7, 17syl21anc 838 . . 3 ((𝜑𝑥𝐴) → (𝐻‘((𝐹𝑥)𝑅(𝐺𝑥))) = ((𝐻‘(𝐹𝑥))𝑆(𝐻‘(𝐺𝑥))))
1918mpteq2dva 5248 . 2 (𝜑 → (𝑥𝐴 ↦ (𝐻‘((𝐹𝑥)𝑅(𝐺𝑥)))) = (𝑥𝐴 ↦ ((𝐻‘(𝐹𝑥))𝑆(𝐻‘(𝐺𝑥)))))
201ffnd 6738 . . . . 5 (𝜑𝐹 Fn 𝐴)
213ffnd 6738 . . . . 5 (𝜑𝐺 Fn 𝐴)
22 coof.a . . . . 5 (𝜑𝐴𝑉)
23 inidm 4235 . . . . 5 (𝐴𝐴) = 𝐴
24 eqidd 2736 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
25 eqidd 2736 . . . . 5 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝐺𝑥))
2620, 21, 22, 22, 23, 24, 25offval 7706 . . . 4 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2726coeq2d 5876 . . 3 (𝜑 → (𝐻 ∘ (𝐹f 𝑅𝐺)) = (𝐻 ∘ (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))))
28 coof.h . . . . 5 (𝜑𝐻 Fn 𝐵)
29 dffn3 6749 . . . . 5 (𝐻 Fn 𝐵𝐻:𝐵⟶ran 𝐻)
3028, 29sylib 218 . . . 4 (𝜑𝐻:𝐵⟶ran 𝐻)
312, 4jca 511 . . . . 5 ((𝜑𝑥𝐴) → ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐵))
32 coof.1 . . . . . 6 ((𝜑 ∧ (𝑏𝐵𝑐𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵)
3332caovclg 7625 . . . . 5 ((𝜑 ∧ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐵)) → ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ 𝐵)
3431, 33syldan 591 . . . 4 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ 𝐵)
3530, 34cofmpt 7152 . . 3 (𝜑 → (𝐻 ∘ (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))) = (𝑥𝐴 ↦ (𝐻‘((𝐹𝑥)𝑅(𝐺𝑥)))))
3627, 35eqtrd 2775 . 2 (𝜑 → (𝐻 ∘ (𝐹f 𝑅𝐺)) = (𝑥𝐴 ↦ (𝐻‘((𝐹𝑥)𝑅(𝐺𝑥)))))
37 fnfco 6774 . . . 4 ((𝐻 Fn 𝐵𝐹:𝐴𝐵) → (𝐻𝐹) Fn 𝐴)
3828, 1, 37syl2anc 584 . . 3 (𝜑 → (𝐻𝐹) Fn 𝐴)
39 fnfco 6774 . . . 4 ((𝐻 Fn 𝐵𝐺:𝐴𝐵) → (𝐻𝐺) Fn 𝐴)
4028, 3, 39syl2anc 584 . . 3 (𝜑 → (𝐻𝐺) Fn 𝐴)
41 fvco2 7006 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐻𝐹)‘𝑥) = (𝐻‘(𝐹𝑥)))
4220, 41sylan 580 . . 3 ((𝜑𝑥𝐴) → ((𝐻𝐹)‘𝑥) = (𝐻‘(𝐹𝑥)))
43 fvco2 7006 . . . 4 ((𝐺 Fn 𝐴𝑥𝐴) → ((𝐻𝐺)‘𝑥) = (𝐻‘(𝐺𝑥)))
4421, 43sylan 580 . . 3 ((𝜑𝑥𝐴) → ((𝐻𝐺)‘𝑥) = (𝐻‘(𝐺𝑥)))
4538, 40, 22, 22, 23, 42, 44offval 7706 . 2 (𝜑 → ((𝐻𝐹) ∘f 𝑆(𝐻𝐺)) = (𝑥𝐴 ↦ ((𝐻‘(𝐹𝑥))𝑆(𝐻‘(𝐺𝑥)))))
4619, 36, 453eqtr4d 2785 1 (𝜑 → (𝐻 ∘ (𝐹f 𝑅𝐺)) = ((𝐻𝐹) ∘f 𝑆(𝐻𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  cmpt 5231  ran crn 5690  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  rhmply1vsca  22408
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