Theorem fsuppco2 8191

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 8192 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		ffun 5961	. . . 4 ⊢ (𝐺:𝐵⟶𝐵 → Fun 𝐺)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐺)
4		fsuppco2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5		ffun 5961	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
7		funco 5842	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
8	3, 6, 7	syl2anc 691	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
9		fsuppco2.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
10	9	fsuppimpd 8165	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11		fco 5971	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
12	1, 4, 11	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
13		eldifi 3694	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
14		fvco3 6185	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
15	4, 13, 14	syl2an 493	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
16		ssid 3587	. . . . . . . 8 ⊢ (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
18		fsuppco2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
19		fsuppco2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
20	4, 17, 18, 19	suppssr 7213	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
21	20	fveq2d 6107	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
22		fsuppco2.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)
23	22	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍)
24	15, 21, 23	3eqtrd 2648	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍)
25	12, 24	suppss 7212	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
26		ssfi 8065	. . 3 ⊢ (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)
27	10, 25, 26	syl2anc 691	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)
28		fsuppco2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
29		fex 6394	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
30	1, 28, 29	syl2anc 691	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
31		fex 6394	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V)
32	4, 18, 31	syl2anc 691	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
33		coexg 7010	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
34	30, 32, 33	syl2anc 691	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
35		isfsupp 8162	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
36	34, 19, 35	syl2anc 691	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
37	8, 27, 36	mpbir2and 959	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540   class class class wbr 4583   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841   finSupp cfsupp 8158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-supp 7183  df-er 7629  df-en 7842  df-fin 7845  df-fsupp 8159

This theorem is referenced by:  gsumzinv  18168  gsumsub  18171