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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 8861 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 6512 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
3 | fsuppco2.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
4 | 3 | ffund 6512 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
5 | funco 6389 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
6 | 2, 4, 5 | syl2anc 586 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
7 | fsuppco2.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
8 | 7 | fsuppimpd 8834 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
9 | fco 6525 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐹:𝐴⟶𝐵) → (𝐺 ∘ 𝐹):𝐴⟶𝐵) | |
10 | 1, 3, 9 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐵) |
11 | eldifi 4102 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
12 | fvco3 6754 | . . . . . 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 3989 | . . . . . . 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 7855 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
18 | 17 | fveq2d 6668 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
19 | fsuppco2.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 𝑍) | |
20 | 19 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 𝑍) |
21 | 13, 18, 20 | 3eqtrd 2860 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 𝑍) |
22 | 10, 21 | suppss 7854 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) |
23 | 8, 22 | ssfid 8735 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin) |
24 | fsuppco2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
25 | fex 6983 | . . . . 5 ⊢ ((𝐺:𝐵⟶𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐺 ∈ V) | |
26 | 1, 24, 25 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
27 | fex 6983 | . . . . 5 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑈) → 𝐹 ∈ V) | |
28 | 3, 15, 27 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
29 | coexg 7628 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
30 | 26, 28, 29 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
31 | isfsupp 8831 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin))) | |
32 | 30, 16, 31 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 𝑍 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 𝑍) ∈ Fin))) |
33 | 6, 23, 32 | mpbir2and 711 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 class class class wbr 5058 ∘ ccom 5553 Fun wfun 6343 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 supp csupp 7824 Fincfn 8503 finSupp cfsupp 8827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-supp 7825 df-er 8283 df-en 8504 df-fin 8507 df-fsupp 8828 |
This theorem is referenced by: gsumzinv 19059 gsumsub 19062 |
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