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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldepsnlinclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for ldepsnlinc 46579. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxzldep.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
zlmodzxzldep.a | ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} |
zlmodzxzldep.b | ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} |
Ref | Expression |
---|---|
ldepsnlinclem2 | ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8787 | . 2 ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤring)) | |
2 | zlmodzxzldep.a | . . . . 5 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
3 | prex 5389 | . . . . 5 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
4 | 2, 3 | eqeltri 2834 | . . . 4 ⊢ 𝐴 ∈ V |
5 | 4 | fsn2 7082 | . . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
6 | oveq1 7364 | . . . . . 6 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹( linC ‘𝑍){𝐴}) = ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴})) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) = ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴})) |
8 | zlmodzxzldep.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
9 | 8 | zlmodzxzlmod 46420 | . . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
10 | 9 | simpli 484 | . . . . . . 7 ⊢ 𝑍 ∈ LMod |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝑍 ∈ LMod) |
12 | 3z 12536 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
13 | 6nn 12242 | . . . . . . . . . 10 ⊢ 6 ∈ ℕ | |
14 | 13 | nnzi 12527 | . . . . . . . . 9 ⊢ 6 ∈ ℤ |
15 | 8 | zlmodzxzel 46421 | . . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍)) |
16 | 12, 14, 15 | mp2an 690 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍) |
17 | 2, 16 | eqeltri 2834 | . . . . . . 7 ⊢ 𝐴 ∈ (Base‘𝑍) |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝐴 ∈ (Base‘𝑍)) |
19 | simpl 483 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹‘𝐴) ∈ (Base‘ℤring)) | |
20 | eqid 2736 | . . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
21 | 9 | simpri 486 | . . . . . . 7 ⊢ ℤring = (Scalar‘𝑍) |
22 | eqid 2736 | . . . . . . 7 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
23 | eqid 2736 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑍) = ( ·𝑠 ‘𝑍) | |
24 | 20, 21, 22, 23 | lincvalsng 46487 | . . . . . 6 ⊢ ((𝑍 ∈ LMod ∧ 𝐴 ∈ (Base‘𝑍) ∧ (𝐹‘𝐴) ∈ (Base‘ℤring)) → ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
25 | 11, 18, 19, 24 | syl3anc 1371 | . . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
26 | 7, 25 | eqtrd 2776 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
27 | eqid 2736 | . . . . . 6 ⊢ {〈0, 0〉, 〈1, 0〉} = {〈0, 0〉, 〈1, 0〉} | |
28 | eqid 2736 | . . . . . 6 ⊢ (-g‘𝑍) = (-g‘𝑍) | |
29 | zlmodzxzldep.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
30 | 8, 27, 23, 28, 2, 29 | zlmodzxznm 46568 | . . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) |
31 | r19.26 3114 | . . . . . 6 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) ↔ (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ ∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴)) | |
32 | oveq1 7364 | . . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) | |
33 | 32 | neeq1d 3003 | . . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
34 | 33 | rspcv 3577 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
35 | zringbas 20875 | . . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring) | |
36 | 35 | eqcomi 2745 | . . . . . . . . . . 11 ⊢ (Base‘ℤring) = ℤ |
37 | 36 | eleq2i 2829 | . . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) ↔ (𝐹‘𝐴) ∈ ℤ) |
38 | 37 | biimpi 215 | . . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) → (𝐹‘𝐴) ∈ ℤ) |
39 | 38 | adantr 481 | . . . . . . . 8 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹‘𝐴) ∈ ℤ) |
40 | 34, 39 | syl11 33 | . . . . . . 7 ⊢ (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 → (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
41 | 40 | adantr 481 | . . . . . 6 ⊢ ((∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ ∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
42 | 31, 41 | sylbi 216 | . . . . 5 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
43 | 30, 42 | ax-mp 5 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵) |
44 | 26, 43 | eqnetrd 3011 | . . 3 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵) |
45 | 5, 44 | sylbi 216 | . 2 ⊢ (𝐹:{𝐴}⟶(Base‘ℤring) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵) |
46 | 1, 45 | syl 17 | 1 ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3445 {csn 4586 {cpr 4588 〈cop 4592 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 0cc0 11051 1c1 11052 2c2 12208 3c3 12209 4c4 12210 6c6 12212 ℤcz 12499 Basecbs 17083 Scalarcsca 17136 ·𝑠 cvsca 17137 -gcsg 18750 LModclmod 20322 ℤringczring 20869 freeLMod cfrlm 21152 linC clinc 46475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-prm 16548 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-0g 17323 df-gsum 17324 df-prds 17329 df-pws 17331 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-cntz 19097 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-lmod 20324 df-lss 20393 df-sra 20633 df-rgmod 20634 df-cnfld 20797 df-zring 20870 df-dsmm 21138 df-frlm 21153 df-linc 46477 |
This theorem is referenced by: ldepsnlinc 46579 |
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