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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 6527 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
3 | fsuppcor.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
4 | 3 | ffund 6527 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
5 | funco 6398 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
6 | 2, 4, 5 | syl2anc 587 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
7 | fsuppcor.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
8 | 7 | fsuppimpd 8970 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
9 | fsuppcor.s | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
10 | 1, 9 | fssresd 6564 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷) |
11 | fco2 6550 | . . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷) | |
12 | 10, 3, 11 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷) |
13 | eldifi 4027 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
14 | fvco3 6788 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) | |
15 | 3, 13, 14 | syl2an 599 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
16 | ssidd 3910 | . . . . . . 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 7916 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
20 | 19 | fveq2d 6699 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹‘𝑥)) = (𝐺‘𝑍)) |
21 | fsuppcor.i | . . . . . 6 ⊢ (𝜑 → (𝐺‘𝑍) = 0 ) | |
22 | 21 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘𝑍) = 0 ) |
23 | 15, 20, 22 | 3eqtrd 2775 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 ) |
24 | 12, 23 | suppss 7914 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) |
25 | 8, 24 | ssfid 8876 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin) |
26 | fsuppcor.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
27 | 1, 26 | fexd 7021 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
28 | 3, 17 | fexd 7021 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
29 | coexg 7685 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
30 | 27, 28, 29 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
31 | fsuppcor.0 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑊) | |
32 | isfsupp 8967 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) | |
33 | 30, 31, 32 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) |
34 | 6, 25, 33 | mpbir2and 713 | 1 ⊢ (𝜑 → (𝐺 ∘ 𝐹) finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∖ cdif 3850 ⊆ wss 3853 class class class wbr 5039 ↾ cres 5538 ∘ ccom 5540 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 supp csupp 7881 Fincfn 8604 finSupp cfsupp 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-supp 7882 df-1o 8180 df-en 8605 df-fin 8608 df-fsupp 8964 |
This theorem is referenced by: mapfienlem1 8999 mapfienlem2 9000 cpmadumatpolylem2 21733 selvval2lem4 39882 |
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