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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 6740 | . . 3 ⊢ (𝜑 → Fun 𝐺) |
| 3 | fsuppcor.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
| 4 | 3 | ffund 6740 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
| 5 | funco 6606 | . . 3 ⊢ ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺 ∘ 𝐹)) | |
| 6 | 2, 4, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → Fun (𝐺 ∘ 𝐹)) |
| 7 | fsuppcor.n | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑍) | |
| 8 | 7 | fsuppimpd 9409 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin) |
| 9 | fsuppcor.s | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) | |
| 10 | 1, 9 | fssresd 6775 | . . . . 5 ⊢ (𝜑 → (𝐺 ↾ 𝐶):𝐶⟶𝐷) |
| 11 | fco2 6762 | . . . . 5 ⊢ (((𝐺 ↾ 𝐶):𝐶⟶𝐷 ∧ 𝐹:𝐴⟶𝐶) → (𝐺 ∘ 𝐹):𝐴⟶𝐷) | |
| 12 | 10, 3, 11 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐺 ∘ 𝐹):𝐴⟶𝐷) |
| 13 | eldifi 4131 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥 ∈ 𝐴) | |
| 14 | fvco3 7008 | . . . . . 6 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝑥 ∈ 𝐴) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) | |
| 15 | 3, 13, 14 | syl2an 596 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = (𝐺‘(𝐹‘𝑥))) |
| 16 | ssidd 4007 | . . . . . . 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 8220 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹‘𝑥) = 𝑍) |
| 20 | 19 | fveq2d 6910 | . . . . 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 2781 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺 ∘ 𝐹)‘𝑥) = 0 ) |
| 24 | 12, 23 | suppss 8219 | . . 3 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) |
| 25 | 8, 24 | ssfid 9301 | . 2 ⊢ (𝜑 → ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin) |
| 26 | fsuppcor.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
| 27 | 1, 26 | fexd 7247 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ V) |
| 28 | 3, 17 | fexd 7247 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ V) |
| 29 | coexg 7951 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺 ∘ 𝐹) ∈ V) | |
| 30 | 27, 28, 29 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ V) |
| 31 | fsuppcor.0 | . . 3 ⊢ (𝜑 → 0 ∈ 𝑊) | |
| 32 | isfsupp 9405 | . . 3 ⊢ (((𝐺 ∘ 𝐹) ∈ V ∧ 0 ∈ 𝑊) → ((𝐺 ∘ 𝐹) finSupp 0 ↔ (Fun (𝐺 ∘ 𝐹) ∧ ((𝐺 ∘ 𝐹) supp 0 ) ∈ Fin))) | |
| 33 | 30, 31, 32 | syl2anc 584 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 ↾ cres 5687 ∘ ccom 5689 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 Fincfn 8985 finSupp cfsupp 9401 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8186 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 |
| This theorem is referenced by: mapfienlem1 9445 mapfienlem2 9446 mhmcompl 22384 cpmadumatpolylem2 22888 selvvvval 42595 |
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