Theorem offsplitfpar 8110

Description: Express the function operation map ∘_f by the functions defined in fsplit 8108 and fpar 8107. (Contributed by AV, 4-Jan-2024.)

Hypotheses

Ref	Expression
fsplitfpar.h	⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
fsplitfpar.s	⊢ 𝑆 = (◡(1st ↾ I ) ↾ 𝐴)

Assertion

Ref	Expression
offsplitfpar	⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))

Proof of Theorem offsplitfpar

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsplitfpar.h	. . . . 5 ⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))
2		fsplitfpar.s	. . . . 5 ⊢ 𝑆 = (◡(1st ↾ I ) ↾ 𝐴)
3	1, 2	fsplitfpar 8109	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
4	3	coeq2d 5846	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
5	4	3ad2ant1 1149	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
6		dffn3 6716	. . . . . 6 ⊢ ( + Fn 𝐶 ↔ + :𝐶⟶ran + )
7	6	birani 508	. . . . 5 ⊢ (( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶) → + :𝐶⟶ran + )
8	7	3ad2ant3 1151	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → + :𝐶⟶ran + )
9		simpl3r 1246	. . . . 5 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (ran 𝐹 × ran 𝐺) ⊆ 𝐶)
10		simp1l 1214	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐹 Fn 𝐴)
11		fnfvelrn 7073	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
12	10, 11	sylan 591	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
13		simp1r 1215	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐺 Fn 𝐴)
14		fnfvelrn 7073	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
15	13, 14	sylan 591	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
16	12, 15	opelxpd 5698	. . . . 5 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (ran 𝐹 × ran 𝐺))
17	9, 16	sseldd 3946	. . . 4 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ 𝐶)
18	8, 17	cofmpt 7126	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
19		df-ov 7411	. . . . 5 ⊢ ((𝐹‘𝑎) + (𝐺‘𝑎)) = ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
20	19	eqcomi 2778	. . . 4 ⊢ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = ((𝐹‘𝑎) + (𝐺‘𝑎))
21	20	mpteq2i 5208	. . 3 ⊢ (𝑎 ∈ 𝐴 ↦ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
22	18, 21	eqtrdi 2820	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
23		offval3 7975	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
24		fndm 6636	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
25		fndm 6636	. . . . . . . 8 ⊢ (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
26	24, 25	ineqan12d 4183	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐴))
27		inidm 4187	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
28	26, 27	eqtrdi 2820	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = 𝐴)
29	28	mpteq1d 5202	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
30	23, 29	sylan9eqr 2826	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
31	30	eqcomd 2775	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝐹 ∘f + 𝐺))
32	31	3adant3 1148	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝐹 ∘f + 𝐺))
33	5, 22, 32	3eqtrd 2808	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ⟨cop 4597   ↦ cmpt 5193   I cid 5553   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   ∘ ccom 5663   Fn wfn 6529  ⟶wf 6530  ‘cfv 6534  (class class class)co 7408   ∘_f cof 7670  1^st c1st 7980  2^nd c2nd 7981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-1st 7982  df-2nd 7983

This theorem is referenced by: (None)