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Mirrors > Home > MPE Home > Th. List > Mathboxes > snlindsntorlem | Structured version Visualization version GIF version |
Description: Lemma for snlindsntor 44880. (Contributed by AV, 15-Apr-2019.) |
Ref | Expression |
---|---|
snlindsntor.b | ⊢ 𝐵 = (Base‘𝑀) |
snlindsntor.r | ⊢ 𝑅 = (Scalar‘𝑀) |
snlindsntor.s | ⊢ 𝑆 = (Base‘𝑅) |
snlindsntor.0 | ⊢ 0 = (0g‘𝑅) |
snlindsntor.z | ⊢ 𝑍 = (0g‘𝑀) |
snlindsntor.t | ⊢ · = ( ·𝑠 ‘𝑀) |
Ref | Expression |
---|---|
snlindsntorlem | ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2799 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉}) | |
2 | fsng 6876 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) | |
3 | 2 | adantll 713 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) |
4 | 1, 3 | mpbird 260 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠}) |
5 | snssi 4701 | . . . . . 6 ⊢ (𝑠 ∈ 𝑆 → {𝑠} ⊆ 𝑆) | |
6 | 5 | adantl 485 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {𝑠} ⊆ 𝑆) |
7 | 4, 6 | fssd 6502 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆) |
8 | snlindsntor.s | . . . . . . 7 ⊢ 𝑆 = (Base‘𝑅) | |
9 | 8 | fvexi 6659 | . . . . . 6 ⊢ 𝑆 ∈ V |
10 | snex 5297 | . . . . . 6 ⊢ {𝑋} ∈ V | |
11 | 9, 10 | pm3.2i 474 | . . . . 5 ⊢ (𝑆 ∈ V ∧ {𝑋} ∈ V) |
12 | elmapg 8402 | . . . . 5 ⊢ ((𝑆 ∈ V ∧ {𝑋} ∈ V) → ({〈𝑋, 𝑠〉} ∈ (𝑆 ↑m {𝑋}) ↔ {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆)) | |
13 | 11, 12 | mp1i 13 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ∈ (𝑆 ↑m {𝑋}) ↔ {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆)) |
14 | 7, 13 | mpbird 260 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} ∈ (𝑆 ↑m {𝑋})) |
15 | oveq1 7142 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓( linC ‘𝑀){𝑋}) = ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋})) | |
16 | 15 | eqeq1d 2800 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 ↔ ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍)) |
17 | fveq1 6644 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓‘𝑋) = ({〈𝑋, 𝑠〉}‘𝑋)) | |
18 | 17 | eqeq1d 2800 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓‘𝑋) = 0 ↔ ({〈𝑋, 𝑠〉}‘𝑋) = 0 )) |
19 | 16, 18 | imbi12d 348 | . . . 4 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) ↔ (({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 → ({〈𝑋, 𝑠〉}‘𝑋) = 0 ))) |
20 | snlindsntor.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀) | |
21 | snlindsntor.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑀) | |
22 | snlindsntor.t | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑀) | |
23 | 20, 21, 8, 22 | lincvalsng 44825 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
24 | 23 | 3expa 1115 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
25 | 24 | eqeq1d 2800 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 ↔ (𝑠 · 𝑋) = 𝑍)) |
26 | fvsng 6919 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) | |
27 | 26 | adantll 713 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) |
28 | 27 | eqeq1d 2800 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉}‘𝑋) = 0 ↔ 𝑠 = 0 )) |
29 | 25, 28 | imbi12d 348 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ((({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 → ({〈𝑋, 𝑠〉}‘𝑋) = 0 ) ↔ ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
30 | 19, 29 | sylan9bbr 514 | . . 3 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) ∧ 𝑓 = {〈𝑋, 𝑠〉}) → (((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) ↔ ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
31 | 14, 30 | rspcdv 3563 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
32 | 31 | ralrimdva 3154 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (∀𝑓 ∈ (𝑆 ↑m {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ∀𝑠 ∈ 𝑆 ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 {csn 4525 〈cop 4531 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 LModclmod 19627 linC clinc 44813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mulg 18217 df-cntz 18439 df-lmod 19629 df-linc 44815 |
This theorem is referenced by: snlindsntor 44880 |
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