![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 9407 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
3 | fsuppinisegfi.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) | |
4 | 3 | snssd 4814 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ (V ∖ { 0 })) |
5 | imass2 6123 | . . . 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 8197 | . . . 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 4006 | . 2 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ⊆ (𝐹 supp 0 )) |
12 | 2, 11 | ssfid 9299 | 1 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 class class class wbr 5148 ◡ccnv 5688 “ cima 5692 (class class class)co 7431 supp csupp 8184 Fincfn 8984 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8185 df-1o 8505 df-en 8985 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: elrspunidl 33436 |
Copyright terms: Public domain | W3C validator |