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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindslinindimp2lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for lindslinindsimp2 47308. (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 8849 | . . . . . 6 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵) | |
2 | 1 | 3ad2ant3 1134 | . . . . 5 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆)) → 𝑓:𝑆⟶𝐵) |
3 | 2 | adantl 481 | . . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝑓:𝑆⟶𝐵) |
4 | difss 4131 | . . . 4 ⊢ (𝑆 ∖ {𝑥}) ⊆ 𝑆 | |
5 | fssres 6757 | . . . 4 ⊢ ((𝑓:𝑆⟶𝐵 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑆) → (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵) | |
6 | 3, 4, 5 | sylancl 585 | . . 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 6708 | . . 3 ⊢ (𝐺:(𝑆 ∖ {𝑥})⟶𝐵 ↔ (𝑓 ↾ (𝑆 ∖ {𝑥})):(𝑆 ∖ {𝑥})⟶𝐵) |
9 | 6, 8 | sylibr 233 | . 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 6905 | . . 3 ⊢ 𝐵 ∈ V |
12 | difexg 5327 | . . . 4 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∖ {𝑥}) ∈ V) | |
13 | 12 | ad2antrr 723 | . . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → (𝑆 ∖ {𝑥}) ∈ V) |
14 | elmapg 8839 | . . 3 ⊢ ((𝐵 ∈ V ∧ (𝑆 ∖ {𝑥}) ∈ V) → (𝐺 ∈ (𝐵 ↑m (𝑆 ∖ {𝑥})) ↔ 𝐺:(𝑆 ∖ {𝑥})⟶𝐵)) | |
15 | 11, 13, 14 | sylancr 586 | . 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 257 | 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 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∖ cdif 3945 ⊆ wss 3948 {csn 4628 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8826 Basecbs 17151 Scalarcsca 17207 0gc0g 17392 invgcminusg 18862 LModclmod 20702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-map 8828 |
This theorem is referenced by: lindslinindimp2lem4 47306 lindslinindsimp2lem5 47307 |
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