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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 9309 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
3 | fsuppinisegfi.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) | |
4 | 3 | snssd 4768 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ (V ∖ { 0 })) |
5 | imass2 6053 | . . . 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 8101 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) | |
10 | 7, 8, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) |
11 | 6, 10 | sstrd 3953 | . 2 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ⊆ (𝐹 supp 0 )) |
12 | 2, 11 | ssfid 9208 | 1 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3444 ∖ cdif 3906 ⊆ wss 3909 {csn 4585 class class class wbr 5104 ◡ccnv 5631 “ cima 5635 (class class class)co 7354 supp csupp 8089 Fincfn 8880 finSupp cfsupp 9302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-supp 8090 df-1o 8409 df-en 8881 df-fin 8884 df-fsupp 9303 |
This theorem is referenced by: elrspunidl 32094 |
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