Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > offinsupp1 | Structured version Visualization version GIF version |
Description: Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.) |
Ref | Expression |
---|---|
offinsupp1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offinsupp1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
offinsupp1.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
offinsupp1.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
offinsupp1.g | ⊢ (𝜑 → 𝐺:𝐴⟶𝑇) |
offinsupp1.1 | ⊢ (𝜑 → 𝐹 finSupp 𝑌) |
offinsupp1.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍) |
Ref | Expression |
---|---|
offinsupp1 | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offinsupp1.1 | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 𝑌) | |
2 | 1 | fsuppimpd 8840 | . . 3 ⊢ (𝜑 → (𝐹 supp 𝑌) ∈ Fin) |
3 | ssidd 3990 | . . . 4 ⊢ (𝜑 → (𝐹 supp 𝑌) ⊆ (𝐹 supp 𝑌)) | |
4 | offinsupp1.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍) | |
5 | offinsupp1.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
6 | offinsupp1.g | . . . 4 ⊢ (𝜑 → 𝐺:𝐴⟶𝑇) | |
7 | offinsupp1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | offinsupp1.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
9 | 3, 4, 5, 6, 7, 8 | suppssof1 7863 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑌)) |
10 | 2, 9 | ssfid 8741 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin) |
11 | ovexd 7191 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑆 ∧ 𝑗 ∈ 𝑇)) → (𝑖𝑅𝑗) ∈ V) | |
12 | inidm 4195 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
13 | 11, 5, 6, 7, 7, 12 | off 7424 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐴⟶V) |
14 | 13 | ffund 6518 | . . 3 ⊢ (𝜑 → Fun (𝐹 ∘f 𝑅𝐺)) |
15 | ovexd 7191 | . . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) ∈ V) | |
16 | offinsupp1.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊) | |
17 | funisfsupp 8838 | . . 3 ⊢ ((Fun (𝐹 ∘f 𝑅𝐺) ∧ (𝐹 ∘f 𝑅𝐺) ∈ V ∧ 𝑍 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin)) | |
18 | 14, 15, 16, 17 | syl3anc 1367 | . 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹 ∘f 𝑅𝐺) supp 𝑍) ∈ Fin)) |
19 | 10, 18 | mpbird 259 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 class class class wbr 5066 Fun wfun 6349 ⟶wf 6351 (class class class)co 7156 ∘f cof 7407 supp csupp 7830 Fincfn 8509 finSupp cfsupp 8833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-supp 7831 df-er 8289 df-en 8510 df-fin 8513 df-fsupp 8834 |
This theorem is referenced by: fedgmullem1 31025 |
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