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Mirrors > Home > MPE Home > Th. List > Mathboxes > snlindsntorlem | Structured version Visualization version GIF version |
Description: Lemma for snlindsntor 48317. (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 2736 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉}) | |
2 | fsng 7157 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) | |
3 | 2 | adantll 714 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠} ↔ {〈𝑋, 𝑠〉} = {〈𝑋, 𝑠〉})) |
4 | 1, 3 | mpbird 257 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶{𝑠}) |
5 | snssi 4813 | . . . . . 6 ⊢ (𝑠 ∈ 𝑆 → {𝑠} ⊆ 𝑆) | |
6 | 5 | adantl 481 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {𝑠} ⊆ 𝑆) |
7 | 4, 6 | fssd 6754 | . . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉}:{𝑋}⟶𝑆) |
8 | snlindsntor.s | . . . . . . 7 ⊢ 𝑆 = (Base‘𝑅) | |
9 | 8 | fvexi 6921 | . . . . . 6 ⊢ 𝑆 ∈ V |
10 | snex 5442 | . . . . . 6 ⊢ {𝑋} ∈ V | |
11 | 9, 10 | pm3.2i 470 | . . . . 5 ⊢ (𝑆 ∈ V ∧ {𝑋} ∈ V) |
12 | elmapg 8878 | . . . . 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 257 | . . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → {〈𝑋, 𝑠〉} ∈ (𝑆 ↑m {𝑋})) |
15 | oveq1 7438 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓( linC ‘𝑀){𝑋}) = ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋})) | |
16 | 15 | eqeq1d 2737 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓( linC ‘𝑀){𝑋}) = 𝑍 ↔ ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍)) |
17 | fveq1 6906 | . . . . . 6 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → (𝑓‘𝑋) = ({〈𝑋, 𝑠〉}‘𝑋)) | |
18 | 17 | eqeq1d 2737 | . . . . 5 ⊢ (𝑓 = {〈𝑋, 𝑠〉} → ((𝑓‘𝑋) = 0 ↔ ({〈𝑋, 𝑠〉}‘𝑋) = 0 )) |
19 | 16, 18 | imbi12d 344 | . . . 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 48262 | . . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
24 | 23 | 3expa 1117 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = (𝑠 · 𝑋)) |
25 | 24 | eqeq1d 2737 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉} ( linC ‘𝑀){𝑋}) = 𝑍 ↔ (𝑠 · 𝑋) = 𝑍)) |
26 | fvsng 7200 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) | |
27 | 26 | adantll 714 | . . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → ({〈𝑋, 𝑠〉}‘𝑋) = 𝑠) |
28 | 27 | eqeq1d 2737 | . . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (({〈𝑋, 𝑠〉}‘𝑋) = 0 ↔ 𝑠 = 0 )) |
29 | 25, 28 | imbi12d 344 | . . . 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 510 | . . 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 3152 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 {csn 4631 〈cop 4637 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LModclmod 20875 linC clinc 48250 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-mulg 19099 df-cntz 19348 df-lmod 20877 df-linc 48252 |
This theorem is referenced by: snlindsntor 48317 |
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