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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindszr | Structured version Visualization version GIF version |
Description: Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.) |
Ref | Expression |
---|---|
lindszr | ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → 𝑆 linIndS 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1055 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ¬ (Scalar‘𝑀) ∈ NzRing) | |
2 | eqid 2610 | . . . . . . 7 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀) | |
3 | 2 | lmodring 18694 | . . . . . 6 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring) |
4 | 3 | 3ad2ant1 1075 | . . . . 5 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Ring) |
5 | 0ringnnzr 19090 | . . . . 5 ⊢ ((Scalar‘𝑀) ∈ Ring → ((#‘(Base‘(Scalar‘𝑀))) = 1 ↔ ¬ (Scalar‘𝑀) ∈ NzRing)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((#‘(Base‘(Scalar‘𝑀))) = 1 ↔ ¬ (Scalar‘𝑀) ∈ NzRing)) |
7 | 1, 6 | mpbird 246 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → (#‘(Base‘(Scalar‘𝑀))) = 1) |
8 | 7 | olcd 407 | . 2 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → ((#‘(Base‘(Scalar‘𝑀))) = 0 ∨ (#‘(Base‘(Scalar‘𝑀))) = 1)) |
9 | eqid 2610 | . . 3 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
10 | eqid 2610 | . . 3 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀)) | |
11 | 9, 2, 10 | lindsrng01 42051 | . 2 ⊢ ((𝑀 ∈ LMod ∧ ((#‘(Base‘(Scalar‘𝑀))) = 0 ∨ (#‘(Base‘(Scalar‘𝑀))) = 1) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → 𝑆 linIndS 𝑀) |
12 | 8, 11 | syld3an2 1365 | 1 ⊢ ((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → 𝑆 linIndS 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 𝒫 cpw 4108 class class class wbr 4583 ‘cfv 5804 0cc0 9815 1c1 9816 #chash 12979 Basecbs 15695 Scalarcsca 15771 Ringcrg 18370 LModclmod 18686 NzRingcnzr 19078 linIndS clininds 42023 |
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-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-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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 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-card 8648 df-cda 8873 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-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-nzr 19079 df-lininds 42025 |
This theorem is referenced by: (None) |
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