Theorem fsuppco2 8712

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 8713 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 6386	. . 3 ⊢ (𝜑 → Fun 𝐺)
3		fsuppco2.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
4	3	ffund 6386	. . 3 ⊢ (𝜑 → Fun 𝐹)
5		funco 6265	. . 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 8686	. . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9		fco 6399	. . . . 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 4024	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴)
12		fvco3 6627	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
13	3, 11, 12	syl2an 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥)))
14		ssidd 3911	. . . . . . 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 7712	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍)
18	17	fveq2d 6542	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍))
19		fsuppco2.i	. . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍)
20	19	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍)
21	13, 18, 20	3eqtrd 2835	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍)
22	10, 21	suppss 7711	. . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
23	8, 22	ssfid 8587	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)
24		fsuppco2.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
25		fex 6855	. . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V)
26	1, 24, 25	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
27		fex 6855	. . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V)
28	3, 15, 27	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
29		coexg 7490	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V)
30	26, 28, 29	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V)
31		isfsupp 8683	. . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
32	30, 16, 31	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin)))
33	6, 23, 32	mpbir2and 709	1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∖ cdif 3856   class class class wbr 4962   ∘ ccom 5447  Fun wfun 6219  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016   supp csupp 7681  Fincfn 8357   finSupp cfsupp 8679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-supp 7682  df-er 8139  df-en 8358  df-fin 8361  df-fsupp 8680

This theorem is referenced by:  gsumzinv  18785  gsumsub  18788