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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindslinindimp2lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for lindslinindsimp2 45431. (Contributed by AV, 25-Apr-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 |
---|---|
lindslinindimp2lem2 | ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8519 | . . . . . 6 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵) | |
2 | 1 | 3ad2ant3 1137 | . . . . 5 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆)) → 𝑓:𝑆⟶𝐵) |
3 | 2 | adantl 485 | . . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝑓:𝑆⟶𝐵) |
4 | difss 4036 | . . . 4 ⊢ (𝑆 ∖ {𝑥}) ⊆ 𝑆 | |
5 | fssres 6574 | . . . 4 ⊢ ((𝑓:𝑆⟶𝐵 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵) | |
6 | 3, 4, 5 | sylancl 589 | . . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵) |
7 | lindslinind.g | . . . 4 ⊢ 𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥})) | |
8 | 7 | feq1i 6525 | . . 3 ⊢ (𝐺:(𝑆 ∖ {𝑥})⟶𝐵 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵) |
9 | 6, 8 | sylibr 237 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝐺:(𝑆 ∖ {𝑥})⟶𝐵) |
10 | lindslinind.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
11 | 10 | fvexi 6720 | . . 3 ⊢ 𝐵 ∈ V |
12 | difexg 5209 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V) | |
13 | 12 | ad2antrr 726 | . . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝑆 ∖ {𝑥}) ∈ V) |
14 | elmapg 8510 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝑆 ∖ {𝑥}) ∈ V) → (𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵)) | |
15 | 11, 13, 14 | sylancr 590 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵)) |
16 | 9, 15 | mpbird 260 | 1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3401 ∖ cdif 3854 ⊆ wss 3857 {csn 4531 ↾ cres 5542 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 Basecbs 16684 Scalarcsca 16770 0gc0g 16916 invgcminusg 18338 LModclmod 19871 |
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-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 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-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 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-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-map 8499 |
This theorem is referenced by: lindslinindimp2lem4 45429 lindslinindsimp2lem5 45430 |
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