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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsuppinisegfi | Structured version Visualization version GIF version |
Description: The initial segment (◡𝐹 “ {𝑌}) of a nonzero 𝑌 is finite if 𝐹 has finite support. (Contributed by Thierry Arnoux, 21-Jun-2024.) |
Ref | Expression |
---|---|
fsuppinisegfi.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
fsuppinisegfi.2 | ⊢ (𝜑 → 0 ∈ 𝑊) |
fsuppinisegfi.3 | ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) |
fsuppinisegfi.4 | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Ref | Expression |
---|---|
fsuppinisegfi | ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsuppinisegfi.4 | . . 3 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
2 | 1 | fsuppimpd 8981 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
3 | fsuppinisegfi.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) | |
4 | 3 | snssd 4712 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ (V ∖ { 0 })) |
5 | imass2 5959 | . . . 4 ⊢ ({𝑌} ⊆ (V ∖ { 0 }) → (◡𝐹 “ {𝑌}) ⊆ (◡𝐹 “ (V ∖ { 0 }))) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
7 | fsuppinisegfi.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
8 | fsuppinisegfi.2 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑊) | |
9 | suppimacnvss 7904 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) | |
10 | 7, 8, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) |
11 | 6, 10 | sstrd 3901 | . 2 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ⊆ (𝐹 supp 0 )) |
12 | 2, 11 | ssfid 8887 | 1 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3401 ∖ cdif 3854 ⊆ wss 3857 {csn 4531 class class class wbr 5043 ◡ccnv 5539 “ cima 5543 (class class class)co 7202 supp csupp 7892 Fincfn 8615 finSupp cfsupp 8974 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
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 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-supp 7893 df-1o 8191 df-en 8616 df-fin 8619 df-fsupp 8975 |
This theorem is referenced by: elrspunidl 31292 |
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