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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindslinindimp2lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for lindslinindsimp2 44525. (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 |
---|---|
lindslinindimp2lem1 | ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝑌 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindslinind.y | . 2 ⊢ 𝑌 = ((invg‘𝑅)‘(𝑓‘𝑥)) | |
2 | lindslinind.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑀) | |
3 | 2 | lmodfgrp 19645 | . . . 4 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Grp) |
4 | 3 | adantl 484 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) → 𝑅 ∈ Grp) |
5 | elmapi 8430 | . . . . . 6 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → 𝑓:𝑆⟶𝐵) | |
6 | ffvelrn 6851 | . . . . . . . 8 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑓‘𝑥) ∈ 𝐵) | |
7 | 6 | a1d 25 | . . . . . . 7 ⊢ ((𝑓:𝑆⟶𝐵 ∧ 𝑥 ∈ 𝑆) → (𝑆 ⊆ (Base‘𝑀) → (𝑓‘𝑥) ∈ 𝐵)) |
8 | 7 | ex 415 | . . . . . 6 ⊢ (𝑓:𝑆⟶𝐵 → (𝑥 ∈ 𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑓‘𝑥) ∈ 𝐵))) |
9 | 5, 8 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐵 ↑m 𝑆) → (𝑥 ∈ 𝑆 → (𝑆 ⊆ (Base‘𝑀) → (𝑓‘𝑥) ∈ 𝐵))) |
10 | 9 | com13 88 | . . . 4 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑥 ∈ 𝑆 → (𝑓 ∈ (𝐵 ↑m 𝑆) → (𝑓‘𝑥) ∈ 𝐵))) |
11 | 10 | 3imp 1107 | . . 3 ⊢ ((𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆)) → (𝑓‘𝑥) ∈ 𝐵) |
12 | lindslinind.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
13 | eqid 2823 | . . . 4 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
14 | 12, 13 | grpinvcl 18153 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑓‘𝑥) ∈ 𝐵) → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ 𝐵) |
15 | 4, 11, 14 | syl2an 597 | . 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → ((invg‘𝑅)‘(𝑓‘𝑥)) ∈ 𝐵) |
16 | 1, 15 | eqeltrid 2919 | 1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥 ∈ 𝑆 ∧ 𝑓 ∈ (𝐵 ↑m 𝑆))) → 𝑌 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 Basecbs 16485 Scalarcsca 16570 0gc0g 16715 Grpcgrp 18105 invgcminusg 18106 LModclmod 19636 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-ring 19301 df-lmod 19638 |
This theorem is referenced by: (None) |
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