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Theorem offsplitfpar 8107
Description: Express the function operation map ∘f by the functions defined in fsplit 8105 and fpar 8104. (Contributed by AV, 4-Jan-2024.)
Hypotheses
Ref Expression
fsplitfpar.h 𝐻 = ((β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) ∩ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))))
fsplitfpar.s 𝑆 = (β—‘(1st β†Ύ I ) β†Ύ 𝐴)
Assertion
Ref Expression
offsplitfpar (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))

Proof of Theorem offsplitfpar
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 fsplitfpar.h . . . . 5 𝐻 = ((β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) ∩ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))))
2 fsplitfpar.s . . . . 5 𝑆 = (β—‘(1st β†Ύ I ) β†Ύ 𝐴)
31, 2fsplitfpar 8106 . . . 4 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (𝐻 ∘ 𝑆) = (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩))
43coeq2d 5861 . . 3 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)))
543ad2ant1 1131 . 2 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)))
6 dffn3 6729 . . . . . . 7 ( + Fn 𝐢 ↔ + :𝐢⟢ran + )
76biimpi 215 . . . . . 6 ( + Fn 𝐢 β†’ + :𝐢⟢ran + )
87adantr 479 . . . . 5 (( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢) β†’ + :𝐢⟢ran + )
983ad2ant3 1133 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ + :𝐢⟢ran + )
10 simpl3r 1227 . . . . 5 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)
11 simp1l 1195 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ 𝐹 Fn 𝐴)
12 fnfvelrn 7081 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜π‘Ž) ∈ ran 𝐹)
1311, 12sylan 578 . . . . . 6 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜π‘Ž) ∈ ran 𝐹)
14 simp1r 1196 . . . . . . 7 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ 𝐺 Fn 𝐴)
15 fnfvelrn 7081 . . . . . . 7 ((𝐺 Fn 𝐴 ∧ π‘Ž ∈ 𝐴) β†’ (πΊβ€˜π‘Ž) ∈ ran 𝐺)
1614, 15sylan 578 . . . . . 6 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ (πΊβ€˜π‘Ž) ∈ ran 𝐺)
1713, 16opelxpd 5714 . . . . 5 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩ ∈ (ran 𝐹 Γ— ran 𝐺))
1810, 17sseldd 3982 . . . 4 ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) ∧ π‘Ž ∈ 𝐴) β†’ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩ ∈ 𝐢)
199, 18cofmpt 7131 . . 3 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)) = (π‘Ž ∈ 𝐴 ↦ ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)))
20 df-ov 7414 . . . . 5 ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž)) = ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)
2120eqcomi 2739 . . . 4 ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩) = ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))
2221mpteq2i 5252 . . 3 (π‘Ž ∈ 𝐴 ↦ ( + β€˜βŸ¨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž)))
2319, 22eqtrdi 2786 . 2 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (π‘Ž ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘Ž), (πΊβ€˜π‘Ž)⟩)) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
24 offval3 7971 . . . . 5 ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) β†’ (𝐹 ∘f + 𝐺) = (π‘Ž ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
25 fndm 6651 . . . . . . . 8 (𝐹 Fn 𝐴 β†’ dom 𝐹 = 𝐴)
26 fndm 6651 . . . . . . . 8 (𝐺 Fn 𝐴 β†’ dom 𝐺 = 𝐴)
2725, 26ineqan12d 4213 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐴))
28 inidm 4217 . . . . . . 7 (𝐴 ∩ 𝐴) = 𝐴
2927, 28eqtrdi 2786 . . . . . 6 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (dom 𝐹 ∩ dom 𝐺) = 𝐴)
3029mpteq1d 5242 . . . . 5 ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (π‘Ž ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
3124, 30sylan9eqr 2792 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š)) β†’ (𝐹 ∘f + 𝐺) = (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))))
3231eqcomd 2736 . . 3 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š)) β†’ (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))) = (𝐹 ∘f + 𝐺))
33323adant3 1130 . 2 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ (π‘Ž ∈ 𝐴 ↦ ((πΉβ€˜π‘Ž) + (πΊβ€˜π‘Ž))) = (𝐹 ∘f + 𝐺))
345, 23, 333eqtrd 2774 1 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  βŸ¨cop 4633   ↦ cmpt 5230   I cid 5572   Γ— cxp 5673  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ∘f cof 7670  1st c1st 7975  2nd c2nd 7976
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-1st 7977  df-2nd 7978
This theorem is referenced by: (None)
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