Theorem fsuppco2 9400

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 9401 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 6717	. . 3 ⊢ (𝜑 → Fun 𝐺)
3		fsuppco2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffund 6717	. . 3 ⊢ (𝜑 → Fun 𝐹)
5		funco 6583	. . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹))
6	2, 4, 5	syl2anc 596	. 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹))
7		fsuppco2.n	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍)
8	7	fsuppimpd 9366	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9		fco 6737	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
10	1, 3, 9	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵)
11		eldifi 4066	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
12		fvco3 6988	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
13	3, 11, 12	syl2an 608	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
14		ssidd 3932	. . . . . . 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 8217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
18	17	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
19		fsuppco2.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)
20	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍)
21	13, 18, 20	3eqtrd 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍)
22	10, 21	suppss 8216	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
23	8, 22	ssfid 9265	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)
24		fsuppco2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25	1, 24	fexd 7235	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
26	3, 15	fexd 7235	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
27		coexg 7943	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
28	25, 26, 27	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
29		isfsupp 9362	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
30	28, 16, 29	syl2anc 596	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
31	6, 23, 30	mpbir2and 726	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∖ cdif 4044   class class class wbr 5102   ∘ ccom 5654  Fun wfun 6535  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422   supp csupp 8181  Fincfn 8978   finSupp cfsupp 9358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-supp 8182  df-1o 8481  df-en 8979  df-fin 8982  df-fsupp 9359

This theorem is used by:  gsumzinv  20166  gsumsub  20169  elrgspnlem1  33851