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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 9396 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 6719 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
3 | fsuppco2.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
4 | 3 | ffund 6719 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
5 | funco 6586 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
6 | 2, 4, 5 | syl2anc 585 | . 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 6739 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵) | |
10 | 1, 3, 9 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵) |
11 | eldifi 4126 | . . . . . 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 597 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
14 | ssidd 4005 | . . . . . . 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 8178 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
18 | 17 | fveq2d 6893 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
19 | fsuppco2.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍) | |
20 | 19 | adantr 482 | . . . . 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 8176 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
23 | 8, 22 | ssfid 9264 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin) |
24 | fsuppco2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
25 | 1, 24 | fexd 7226 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
26 | 3, 15 | fexd 7226 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
27 | coexg 7917 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
28 | 25, 26, 27 | syl2anc 585 | . . 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 585 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin))) |
31 | 6, 23, 30 | mpbir2and 712 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3945 class class class wbr 5148 ∘ ccom 5680 Fun wfun 6535 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 supp csupp 8143 Fincfn 8936 finSupp cfsupp 9358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-supp 8144 df-1o 8463 df-en 8937 df-fin 8940 df-fsupp 9359 |
This theorem is referenced by: gsumzinv 19808 gsumsub 19811 |
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