Theorem fsuppco2 9304

Description: The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 9305 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
fsuppco2.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
fsuppco2.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fsuppco2.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
fsuppco2.a	⊢ (𝜑 → 𝐴 ∈ 𝑈)
fsuppco2.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fsuppco2.n	⊢ (𝜑 → 𝐹 finSupp 𝑍)
fsuppco2.i	⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)

Assertion

Ref	Expression
fsuppco2	⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)

Proof of Theorem fsuppco2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppco2.g	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵)
2	1	ffund 6664	. . 3 ⊢ (𝜑 → Fun 𝐺)
3		fsuppco2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffund 6664	. . 3 ⊢ (𝜑 → Fun 𝐹)
5		funco 6530	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
6	2, 4, 5	syl2anc 584	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
7		fsuppco2.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
8	7	fsuppimpd 9270	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9		fco 6684	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
10	1, 3, 9	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
11		eldifi 4081	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
12		fvco3 6931	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
13	3, 11, 12	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
14		ssidd 3955	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
15		fsuppco2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
16		fsuppco2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
17	3, 14, 15, 16	suppssr 8135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
18	17	fveq2d 6836	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
19		fsuppco2.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍)
21	13, 18, 20	3eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍)
22	10, 21	suppss 8134	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
23	8, 22	ssfid 9167	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)
24		fsuppco2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25	1, 24	fexd 7171	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
26	3, 15	fexd 7171	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
27		coexg 7869	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
28	25, 26, 27	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
29		isfsupp 9266	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
30	28, 16, 29	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
31	6, 23, 30	mpbir2and 713	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∖ cdif 3896   class class class wbr 5096   ∘ ccom 5626  Fun wfun 6484  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   supp csupp 8100  Fincfn 8881   finSupp cfsupp 9262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-supp 8101  df-1o 8395  df-en 8882  df-fin 8885  df-fsupp 9263

This theorem is referenced by:  gsumzinv  19872  gsumsub  19875  elrgspnlem1  33273