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Mirrors > Home > MPE Home > Th. List > Mathboxes > snlindsntorlem | Structured version Visualization version GIF version |
Description: Lemma for snlindsntor 44520. (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 2822 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉}) | |
2 | fsng 6893 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) | |
3 | 2 | adantll 712 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) |
4 | 1, 3 | mpbird 259 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠}) |
5 | snssi 4734 | . . . . . 6 ⊢ (𝑠 ∈ 𝑆 → {𝑠} ⊆ 𝑆) | |
6 | 5 | adantl 484 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {𝑠} ⊆ 𝑆) |
7 | 4, 6 | fssd 6522 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆) |
8 | snlindsntor.s | . . . . . . 7 ⊢ 𝑆 = (Base‘𝑅) | |
9 | 8 | fvexi 6678 | . . . . . 6 ⊢ 𝑆 ∈ V |
10 | snex 5323 | . . . . . 6 ⊢ {𝑋} ∈ V | |
11 | 9, 10 | pm3.2i 473 | . . . . 5 ⊢ (𝑆 ∈ V ∧ {𝑋} ∈ V) |
12 | elmapg 8413 | . . . . 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 259 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} ∈ (𝑆 ↑m {𝑋})) |
15 | oveq1 7157 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓( linC ‘𝑀){𝑋}) = ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋})) | |
16 | 15 | eqeq1d 2823 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 ↔ ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍)) |
17 | fveq1 6663 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓‘𝑋) = ({〈𝑋, 𝑠〉}‘𝑋)) | |
18 | 17 | eqeq1d 2823 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓‘𝑋) = 0 ↔ ({〈𝑋, 𝑠〉}‘𝑋) = 0 )) |
19 | 16, 18 | imbi12d 347 | . . . 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 44465 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
24 | 23 | 3expa 1114 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
25 | 24 | eqeq1d 2823 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 ↔ (𝑠 · 𝑋) = 𝑍)) |
26 | fvsng 6936 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) | |
27 | 26 | adantll 712 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) |
28 | 27 | eqeq1d 2823 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉}‘𝑋) = 0 ↔ 𝑠 = 0 )) |
29 | 25, 28 | imbi12d 347 | . . . 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 513 | . . 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 3614 | . 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 3189 | 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 {csn 4560 〈cop 4566 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 Basecbs 16477 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 LModclmod 19628 linC clinc 44453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-0g 16709 df-gsum 16710 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mulg 18219 df-cntz 18441 df-lmod 19630 df-linc 44455 |
This theorem is referenced by: snlindsntor 44520 |
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