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Mirrors > Home > MPE Home > Th. List > fsuppcor | Structured version Visualization version GIF version |
Description: The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.) |
Ref | Expression |
---|---|
fsuppcor.0 | ⊢ (𝜑 → 0 ∈ 𝑊) |
fsuppcor.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
fsuppcor.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
fsuppcor.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) |
fsuppcor.s | ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
fsuppcor.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
fsuppcor.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
fsuppcor.n | ⊢ (𝜑 → 𝐹 finSupp 𝑍) |
fsuppcor.i | ⊢ (𝜑 → (𝐺‘𝑍) = 0 ) |
Ref | Expression |
---|---|
fsuppcor | ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppcor.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐷) | |
2 | 1 | ffund 6734 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
3 | fsuppcor.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
4 | 3 | ffund 6734 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
5 | funco 6601 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
6 | 2, 4, 5 | syl2anc 582 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
7 | fsuppcor.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
8 | 7 | fsuppimpd 9415 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
9 | fsuppcor.s | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
10 | 1, 9 | fssresd 6771 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷) |
11 | fco2 6757 | . . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷) | |
12 | 10, 3, 11 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷) |
13 | eldifi 4126 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
14 | fvco3 7003 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) | |
15 | 3, 13, 14 | syl2an 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
16 | ssidd 4003 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)) | |
17 | fsuppcor.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
18 | fsuppcor.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
19 | 3, 16, 17, 18 | suppssr 8212 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
20 | 19 | fveq2d 6907 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
21 | fsuppcor.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 ) | |
22 | 21 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 0 ) |
23 | 15, 20, 22 | 3eqtrd 2770 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 ) |
24 | 12, 23 | suppss 8210 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) |
25 | 8, 24 | ssfid 9303 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin) |
26 | fsuppcor.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
27 | 1, 26 | fexd 7246 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
28 | 3, 17 | fexd 7246 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
29 | coexg 7944 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
30 | 27, 28, 29 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
31 | fsuppcor.0 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑊) | |
32 | isfsupp 9411 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) | |
33 | 30, 31, 32 | syl2anc 582 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) |
34 | 6, 25, 33 | mpbir2and 711 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∖ cdif 3944 ⊆ wss 3947 class class class wbr 5155 ↾ cres 5686 ∘ ccom 5688 Fun wfun 6550 ⟶wf 6552 ‘cfv 6556 (class class class)co 7426 supp csupp 8176 Fincfn 8976 finSupp cfsupp 9407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-supp 8177 df-1o 8498 df-en 8977 df-fin 8980 df-fsupp 9408 |
This theorem is referenced by: mapfienlem1 9450 mapfienlem2 9451 mhmcompl 22374 cpmadumatpolylem2 22878 selvvvval 42255 |
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