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Mirrors > Home > MPE Home > Th. List > Mathboxes > snlindsntorlem | Structured version Visualization version GIF version |
Description: Lemma for snlindsntor 42260. (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 2623 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉}) | |
2 | fsng 6404 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) | |
3 | 2 | adantll 750 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) |
4 | 1, 3 | mpbird 247 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠}) |
5 | snssi 4339 | . . . . . 6 ⊢ (𝑠 ∈ 𝑆 → {𝑠} ⊆ 𝑆) | |
6 | 5 | adantl 482 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {𝑠} ⊆ 𝑆) |
7 | 4, 6 | fssd 6057 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆) |
8 | snlindsntor.s | . . . . . . 7 ⊢ 𝑆 = (Base‘𝑅) | |
9 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
10 | 8, 9 | eqeltri 2697 | . . . . . 6 ⊢ 𝑆 ∈ V |
11 | snex 4908 | . . . . . 6 ⊢ {𝑋} ∈ V | |
12 | 10, 11 | pm3.2i 471 | . . . . 5 ⊢ (𝑆 ∈ V ∧ {𝑋} ∈ V) |
13 | elmapg 7870 | . . . . 5 ⊢ ((𝑆 ∈ V ∧ {𝑋} ∈ V) → ({〈𝑋, 𝑠〉} ∈ (𝑆 ↑𝑚 {𝑋}) ↔ {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆)) | |
14 | 12, 13 | mp1i 13 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ∈ (𝑆 ↑𝑚 {𝑋}) ↔ {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆)) |
15 | 7, 14 | mpbird 247 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} ∈ (𝑆 ↑𝑚 {𝑋})) |
16 | oveq1 6657 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓( linC ‘𝑀){𝑋}) = ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋})) | |
17 | 16 | eqeq1d 2624 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 ↔ ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍)) |
18 | fveq1 6190 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓‘𝑋) = ({〈𝑋, 𝑠〉}‘𝑋)) | |
19 | 18 | eqeq1d 2624 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓‘𝑋) = 0 ↔ ({〈𝑋, 𝑠〉}‘𝑋) = 0 )) |
20 | 17, 19 | imbi12d 334 | . . . 4 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) ↔ (({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 → ({〈𝑋, 𝑠〉}‘𝑋) = 0 ))) |
21 | snlindsntor.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀) | |
22 | snlindsntor.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑀) | |
23 | snlindsntor.t | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑀) | |
24 | 21, 22, 8, 23 | lincvalsng 42205 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
25 | 24 | 3expa 1265 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
26 | 25 | eqeq1d 2624 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 ↔ (𝑠 · 𝑋) = 𝑍)) |
27 | fvsng 6447 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) | |
28 | 27 | adantll 750 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) |
29 | 28 | eqeq1d 2624 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉}‘𝑋) = 0 ↔ 𝑠 = 0 )) |
30 | 26, 29 | imbi12d 334 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ((({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 → ({〈𝑋, 𝑠〉}‘𝑋) = 0 ) ↔ ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
31 | 20, 30 | sylan9bbr 737 | . . 3 ⊢ ((((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) ∧ 𝑓 = {〈𝑋, 𝑠〉}) → (((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) ↔ ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
32 | 15, 31 | rspcdv 3312 | . 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (∀𝑓 ∈ (𝑆 ↑𝑚 {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓‘𝑋) = 0 ) → ((𝑠 · 𝑋) = 𝑍 → 𝑠 = 0 ))) |
33 | 32 | ralrimdva 2969 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 {csn 4177 〈cop 4183 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 Basecbs 15857 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 LModclmod 18863 linC clinc 42193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-mulg 17541 df-cntz 17750 df-lmod 18865 df-linc 42195 |
This theorem is referenced by: snlindsntor 42260 |
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