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Mirrors > Home > MPE Home > Th. List > Mathboxes > islininds2 | Structured version Visualization version GIF version |
Description: Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
islindeps2.b | ⊢ 𝐵 = (Base‘𝑀) |
islindeps2.z | ⊢ 𝑍 = (0g‘𝑀) |
islindeps2.r | ⊢ 𝑅 = (Scalar‘𝑀) |
islindeps2.e | ⊢ 𝐸 = (Base‘𝑅) |
islindeps2.0 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
islininds2 | ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (𝑆 linIndS 𝑀 → ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindepsnlininds 44500 | . . . . 5 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀)) | |
2 | 1 | ancoms 461 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀)) |
3 | 2 | 3adant3 1128 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀)) |
4 | 3 | con2bid 357 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (𝑆 linIndS 𝑀 ↔ ¬ 𝑆 linDepS 𝑀)) |
5 | notnotb 317 | . . . . . . . . . 10 ⊢ (𝑓 finSupp 0 ↔ ¬ ¬ 𝑓 finSupp 0 ) | |
6 | nne 3020 | . . . . . . . . . . 11 ⊢ (¬ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠 ↔ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) | |
7 | 6 | bicomi 226 | . . . . . . . . . 10 ⊢ ((𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠 ↔ ¬ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠) |
8 | 5, 7 | anbi12i 628 | . . . . . . . . 9 ⊢ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ (¬ ¬ 𝑓 finSupp 0 ∧ ¬ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) |
9 | pm4.56 985 | . . . . . . . . 9 ⊢ ((¬ ¬ 𝑓 finSupp 0 ∧ ¬ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠) ↔ ¬ (¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) | |
10 | 8, 9 | bitri 277 | . . . . . . . 8 ⊢ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ ¬ (¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) |
11 | 10 | rexbii 3247 | . . . . . . 7 ⊢ (∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠})) ¬ (¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) |
12 | rexnal 3238 | . . . . . . 7 ⊢ (∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠})) ¬ (¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠) ↔ ¬ ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) | |
13 | 11, 12 | bitri 277 | . . . . . 6 ⊢ (∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ ¬ ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) |
14 | 13 | rexbii 3247 | . . . . 5 ⊢ (∃𝑠 ∈ 𝑆 ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ ∃𝑠 ∈ 𝑆 ¬ ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) |
15 | rexnal 3238 | . . . . 5 ⊢ (∃𝑠 ∈ 𝑆 ¬ ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠) ↔ ¬ ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) | |
16 | 14, 15 | bitri 277 | . . . 4 ⊢ (∃𝑠 ∈ 𝑆 ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ ¬ ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)) |
17 | islindeps2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
18 | islindeps2.z | . . . . 5 ⊢ 𝑍 = (0g‘𝑀) | |
19 | islindeps2.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑀) | |
20 | islindeps2.e | . . . . 5 ⊢ 𝐸 = (Base‘𝑅) | |
21 | islindeps2.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
22 | 17, 18, 19, 20, 21 | islindeps2 44531 | . . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (∃𝑠 ∈ 𝑆 ∃𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) → 𝑆 linDepS 𝑀)) |
23 | 16, 22 | syl5bir 245 | . . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (¬ ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠) → 𝑆 linDepS 𝑀)) |
24 | 23 | con1d 147 | . 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (¬ 𝑆 linDepS 𝑀 → ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠))) |
25 | 4, 24 | sylbid 242 | 1 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵 ∧ 𝑅 ∈ NzRing) → (𝑆 linIndS 𝑀 → ∀𝑠 ∈ 𝑆 ∀𝑓 ∈ (𝐸 ↑m (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∖ cdif 3933 𝒫 cpw 4539 {csn 4561 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 finSupp cfsupp 8827 Basecbs 16477 Scalarcsca 16562 0gc0g 16707 LModclmod 19628 NzRingcnzr 20024 linC clinc 44452 linIndS clininds 44488 linDepS clindeps 44489 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 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-fsupp 8828 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-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-lmod 19630 df-nzr 20025 df-linc 44454 df-lininds 44490 df-lindeps 44492 |
This theorem is referenced by: (None) |
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