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Mirrors > Home > MPE Home > Th. List > Mathboxes > snlindsntorlem | Structured version Visualization version GIF version |
Description: Lemma for snlindsntor 42054. (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 2611 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉}) | |
2 | fsng 6310 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) | |
3 | 2 | adantll 746 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) |
4 | 1, 3 | mpbird 246 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠}) |
5 | snssi 4280 | . . . . . 6 ⊢ (𝑠 ∈ 𝑆 → {𝑠} ⊆ 𝑆) | |
6 | 5 | adantl 481 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {𝑠} ⊆ 𝑆) |
7 | 4, 6 | fssd 5970 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆) |
8 | snlindsntor.s | . . . . . . 7 ⊢ 𝑆 = (Base‘𝑅) | |
9 | fvex 6113 | . . . . . . 7 ⊢ (Base‘𝑅) ∈ V | |
10 | 8, 9 | eqeltri 2684 | . . . . . 6 ⊢ 𝑆 ∈ V |
11 | snex 4835 | . . . . . 6 ⊢ {𝑋} ∈ V | |
12 | 10, 11 | pm3.2i 470 | . . . . 5 ⊢ (𝑆 ∈ V ∧ {𝑋} ∈ V) |
13 | elmapg 7757 | . . . . 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 246 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} ∈ (𝑆 ↑𝑚 {𝑋})) |
16 | oveq1 6556 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓( linC ‘𝑀){𝑋}) = ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋})) | |
17 | 16 | eqeq1d 2612 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 ↔ ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍)) |
18 | fveq1 6102 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓‘𝑋) = ({〈𝑋, 𝑠〉}‘𝑋)) | |
19 | 18 | eqeq1d 2612 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓‘𝑋) = 0 ↔ ({〈𝑋, 𝑠〉}‘𝑋) = 0 )) |
20 | 17, 19 | imbi12d 333 | . . . 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 41999 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
25 | 24 | 3expa 1257 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
26 | 25 | eqeq1d 2612 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 ↔ (𝑠 · 𝑋) = 𝑍)) |
27 | fvsng 6352 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) | |
28 | 27 | adantll 746 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) |
29 | 28 | eqeq1d 2612 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉}‘𝑋) = 0 ↔ 𝑠 = 0 )) |
30 | 26, 29 | imbi12d 333 | . . . 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 733 | . . 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 3285 | . 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 2952 | 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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 {csn 4125 〈cop 4131 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 LModclmod 18686 linC clinc 41987 |
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 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
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-nel 2783 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-se 4998 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-isom 5813 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-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mulg 17364 df-cntz 17573 df-lmod 18688 df-linc 41989 |
This theorem is referenced by: snlindsntor 42054 |
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