Theorem offsplitfpar 8101

Description: Express the function operation map ∘_f by the functions defined in fsplit 8099 and fpar 8098. (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 8100	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐻 ∘ 𝑆) = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
4	3	coeq2d 5829	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ( + ∘ (𝐻 ∘ 𝑆)) = ( + ∘ (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)))
5	4	3ad2ant1 1133	. 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 6703	. . . . . . 7 ⊢ ( + Fn 𝐶 ↔ + :𝐶⟶ran + )
7	6	biimpi 216	. . . . . 6 ⊢ ( + Fn 𝐶 → + :𝐶⟶ran + )
8	7	adantr 480	. . . . 5 ⊢ (( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶) → + :𝐶⟶ran + )
9	8	3ad2ant3 1135	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → + :𝐶⟶ran + )
10		simpl3r 1230	. . . . 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)
11		simp1l 1198	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐹 Fn 𝐴)
12		fnfvelrn 7055	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
13	11, 12	sylan 580	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ ran 𝐹)
14		simp1r 1199	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → 𝐺 Fn 𝐴)
15		fnfvelrn 7055	. . . . . . 7 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
16	14, 15	sylan 580	. . . . . 6 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ ran 𝐺)
17	13, 16	opelxpd 5680	. . . . 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))
18	10, 17	sseldd 3950	. . . 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)
19	9, 18	cofmpt 7107	. . 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)⟩)))
20		df-ov 7393	. . . . 5 ⊢ ((𝐹‘𝑎) + (𝐺‘𝑎)) = ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
21	20	eqcomi 2739	. . . 4 ⊢ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = ((𝐹‘𝑎) + (𝐺‘𝑎))
22	21	mpteq2i 5206	. . 3 ⊢ (𝑎 ∈ 𝐴 ↦ ( + ‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎)))
23	19, 22	eqtrdi 2781	. 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))))
24		offval3 7964	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
25		fndm 6624	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
26		fndm 6624	. . . . . . . 8 ⊢ (𝐺 Fn 𝐴 → dom 𝐺 = 𝐴)
27	25, 26	ineqan12d 4188	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐴))
28		inidm 4193	. . . . . . 7 ⊢ (𝐴 ∩ 𝐴) = 𝐴
29	27, 28	eqtrdi 2781	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom 𝐹 ∩ dom 𝐺) = 𝐴)
30	29	mpteq1d 5200	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝑎 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
31	24, 30	sylan9eqr 2787	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝐹 ∘f + 𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))))
32	31	eqcomd 2736	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊)) → (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎) + (𝐺‘𝑎))) = (𝐹 ∘f + 𝐺))
33	32	3adant3 1132	. 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))
34	5, 23, 33	3eqtrd 2769	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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  ⟨cop 4598   ↦ cmpt 5191   I cid 5535   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   ↾ cres 5643   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654  1^st c1st 7969  2^nd c2nd 7970

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-1st 7971  df-2nd 7972

This theorem is referenced by: (None)