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Mathbox for Thierry Arnoux |
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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 8858 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
3 | fsuppinisegfi.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) | |
4 | 3 | snssd 4692 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ (V ∖ { 0 })) |
5 | imass2 5930 | . . . 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 7840 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) | |
10 | 7, 8, 9 | syl2anc 588 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) |
11 | 6, 10 | sstrd 3898 | . 2 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ⊆ (𝐹 supp 0 )) |
12 | 2, 11 | ssfid 8755 | 1 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 Vcvv 3407 ∖ cdif 3851 ⊆ wss 3854 {csn 4515 class class class wbr 5025 ◡ccnv 5516 “ cima 5520 (class class class)co 7143 supp csupp 7828 Fincfn 8520 finSupp cfsupp 8851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-supp 7829 df-er 8292 df-en 8521 df-fin 8524 df-fsupp 8852 |
This theorem is referenced by: elrspunidl 31112 |
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