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Mirrors > Home > MPE Home > Th. List > fsuppco2 | Structured version Visualization version GIF version |
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 8942 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.) |
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) |
Ref | Expression |
---|---|
fsuppco2 | ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppco2.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐵) | |
2 | 1 | ffund 6509 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
3 | fsuppco2.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
4 | 3 | ffund 6509 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
5 | funco 6380 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
6 | 2, 4, 5 | syl2anc 587 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
7 | fsuppco2.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
8 | 7 | fsuppimpd 8914 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
9 | fco 6529 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵) | |
10 | 1, 3, 9 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵) |
11 | eldifi 4018 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
12 | fvco3 6768 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) | |
13 | 3, 11, 12 | syl2an 599 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
14 | ssidd 3901 | . . . . . . 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 7892 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
18 | 17 | fveq2d 6679 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
19 | fsuppco2.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍) | |
20 | 19 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍) |
21 | 13, 18, 20 | 3eqtrd 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍) |
22 | 10, 21 | suppss 7890 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
23 | 8, 22 | ssfid 8820 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin) |
24 | fsuppco2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
25 | 1, 24 | fexd 7001 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
26 | 3, 15 | fexd 7001 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
27 | coexg 7661 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
28 | 25, 26, 27 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
29 | isfsupp 8911 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin))) | |
30 | 28, 16, 29 | syl2anc 587 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 Vcvv 3398 ∖ cdif 3841 class class class wbr 5031 ∘ ccom 5530 Fun wfun 6334 ⟶wf 6336 ‘cfv 6340 (class class class)co 7171 supp csupp 7857 Fincfn 8556 finSupp cfsupp 8907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-supp 7858 df-1o 8132 df-en 8557 df-fin 8560 df-fsupp 8908 |
This theorem is referenced by: gsumzinv 19185 gsumsub 19188 |
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