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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindslinindimp2lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for lindslinindsimp2 44517. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
lindslinind.r | ⊢ 𝑅 = (Scalar‘𝑀) |
lindslinind.b | ⊢ 𝐵 = (Base‘𝑅) |
lindslinind.0 | ⊢ 0 = (0g‘𝑅) |
lindslinind.z | ⊢ 𝑍 = (0g‘𝑀) |
lindslinind.y | ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥)) |
lindslinind.g | ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})) |
Ref | Expression |
---|---|
lindslinindimp2lem3 | ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindslinind.g | . 2 ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})) | |
2 | simp3r 1198 | . . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝑓 finSupp 0 ) | |
3 | lindslinind.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
4 | 3 | fvexi 6683 | . . . 4 ⊢ 0 ∈ V |
5 | 4 | a1i 11 | . . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 )) → 0 ∈ V) |
6 | 2, 5 | fsuppres 8857 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 )) → (𝑓 ↾ (𝑆 ∖ {𝑥})) finSupp 0 ) |
7 | 1, 6 | eqbrtrid 5100 | 1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆) ∧ (𝑓 ∈ (𝐵 ↑m 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 {csn 4566 class class class wbr 5065 ↾ cres 5556 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 finSupp cfsupp 8832 Basecbs 16482 Scalarcsca 16567 0gc0g 16712 invgcminusg 18103 LModclmod 19633 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-supp 7830 df-er 8288 df-en 8509 df-fin 8512 df-fsupp 8833 |
This theorem is referenced by: lindslinindimp2lem4 44515 |
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