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Mirrors > Home > MPE Home > Th. List > Mathboxes > lincext2 | Structured version Visualization version GIF version |
Description: Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
lincext.b | ⊢ 𝐵 = (Base‘𝑀) |
lincext.r | ⊢ 𝑅 = (Scalar‘𝑀) |
lincext.e | ⊢ 𝐸 = (Base‘𝑅) |
lincext.0 | ⊢ 0 = (0g‘𝑅) |
lincext.z | ⊢ 𝑍 = (0g‘𝑀) |
lincext.n | ⊢ 𝑁 = (invg‘𝑅) |
lincext.f | ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧))) |
Ref | Expression |
---|---|
lincext2 | ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6113 | . . . . . 6 ⊢ (𝑁‘𝑌) ∈ V | |
2 | fvex 6113 | . . . . . 6 ⊢ (𝐺‘𝑧) ∈ V | |
3 | 1, 2 | ifex 4106 | . . . . 5 ⊢ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧)) ∈ V |
4 | lincext.f | . . . . 5 ⊢ 𝐹 = (𝑧 ∈ 𝑆 ↦ if(𝑧 = 𝑋, (𝑁‘𝑌), (𝐺‘𝑧))) | |
5 | 3, 4 | dmmpti 5936 | . . . 4 ⊢ dom 𝐹 = 𝑆 |
6 | 5 | difeq1i 3686 | . . 3 ⊢ (dom 𝐹 ∖ (𝑆 ∖ {𝑋})) = (𝑆 ∖ (𝑆 ∖ {𝑋})) |
7 | snssi 4280 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑆 → {𝑋} ⊆ 𝑆) | |
8 | 7 | 3ad2ant2 1076 | . . . . . 6 ⊢ ((𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) → {𝑋} ⊆ 𝑆) |
9 | 8 | 3ad2ant2 1076 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → {𝑋} ⊆ 𝑆) |
10 | dfss4 3820 | . . . . 5 ⊢ ({𝑋} ⊆ 𝑆 ↔ (𝑆 ∖ (𝑆 ∖ {𝑋})) = {𝑋}) | |
11 | 9, 10 | sylib 207 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → (𝑆 ∖ (𝑆 ∖ {𝑋})) = {𝑋}) |
12 | snfi 7923 | . . . 4 ⊢ {𝑋} ∈ Fin | |
13 | 11, 12 | syl6eqel 2696 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → (𝑆 ∖ (𝑆 ∖ {𝑋})) ∈ Fin) |
14 | 6, 13 | syl5eqel 2692 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → (dom 𝐹 ∖ (𝑆 ∖ {𝑋})) ∈ Fin) |
15 | lincext.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
16 | lincext.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑀) | |
17 | lincext.e | . . . 4 ⊢ 𝐸 = (Base‘𝑅) | |
18 | lincext.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
19 | lincext.z | . . . 4 ⊢ 𝑍 = (0g‘𝑀) | |
20 | lincext.n | . . . 4 ⊢ 𝑁 = (invg‘𝑅) | |
21 | 15, 16, 17, 18, 19, 20, 4 | lincext1 42037 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸 ↑𝑚 𝑆)) |
22 | 21 | 3adant3 1074 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 ∈ (𝐸 ↑𝑚 𝑆)) |
23 | elmapfun 7767 | . . 3 ⊢ (𝐹 ∈ (𝐸 ↑𝑚 𝑆) → Fun 𝐹) | |
24 | 22, 23 | syl 17 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → Fun 𝐹) |
25 | elmapi 7765 | . . . . 5 ⊢ (𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋})) → 𝐺:(𝑆 ∖ {𝑋})⟶𝐸) | |
26 | 4 | fdmdifeqresdif 41913 | . . . . 5 ⊢ (𝐺:(𝑆 ∖ {𝑋})⟶𝐸 → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
27 | 25, 26 | syl 17 | . . . 4 ⊢ (𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋})) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
28 | 27 | 3ad2ant3 1077 | . . 3 ⊢ ((𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
29 | 28 | 3ad2ant2 1076 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐺 = (𝐹 ↾ (𝑆 ∖ {𝑋}))) |
30 | simp3 1056 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐺 finSupp 0 ) | |
31 | fvex 6113 | . . . 4 ⊢ (0g‘𝑅) ∈ V | |
32 | 18, 31 | eqeltri 2684 | . . 3 ⊢ 0 ∈ V |
33 | 32 | a1i 11 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 0 ∈ V) |
34 | 14, 22, 24, 29, 30, 33 | resfsupp 8185 | 1 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌 ∈ 𝐸 ∧ 𝑋 ∈ 𝑆 ∧ 𝐺 ∈ (𝐸 ↑𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ifcif 4036 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Fincfn 7841 finSupp cfsupp 8158 Basecbs 15695 Scalarcsca 15771 0gc0g 15923 invgcminusg 17246 LModclmod 18686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-fin 7845 df-fsupp 8159 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-ring 18372 df-lmod 18688 |
This theorem is referenced by: lincext3 42039 lindslinindsimp1 42040 islindeps2 42066 |
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