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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 6715 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
3 | fsuppcor.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
4 | 3 | ffund 6715 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
5 | funco 6582 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
6 | 2, 4, 5 | syl2anc 583 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
7 | fsuppcor.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
8 | 7 | fsuppimpd 9371 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
9 | fsuppcor.s | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
10 | 1, 9 | fssresd 6752 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷) |
11 | fco2 6738 | . . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷) | |
12 | 10, 3, 11 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷) |
13 | eldifi 4121 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
14 | fvco3 6984 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) | |
15 | 3, 13, 14 | syl2an 595 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
16 | ssidd 4000 | . . . . . . 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 8181 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
20 | 19 | fveq2d 6889 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
21 | fsuppcor.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 ) | |
22 | 21 | adantr 480 | . . . . 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 8179 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) |
25 | 8, 24 | ssfid 9269 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin) |
26 | fsuppcor.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
27 | 1, 26 | fexd 7224 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
28 | 3, 17 | fexd 7224 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
29 | coexg 7919 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
30 | 27, 28, 29 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
31 | fsuppcor.0 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑊) | |
32 | isfsupp 9367 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) | |
33 | 30, 31, 32 | syl2anc 583 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) |
34 | 6, 25, 33 | mpbir2and 710 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 class class class wbr 5141 ↾ cres 5671 ∘ ccom 5673 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 supp csupp 8146 Fincfn 8941 finSupp cfsupp 9363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-supp 8147 df-1o 8467 df-en 8942 df-fin 8945 df-fsupp 9364 |
This theorem is referenced by: mapfienlem1 9402 mapfienlem2 9403 cpmadumatpolylem2 22739 mhmcompl 41677 selvvvval 41714 |
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