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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldepsnlinclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for ldepsnlinc 44917. (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 8411 | . 2 ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤring)) | |
2 | zlmodzxzldep.a | . . . . 5 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
3 | prex 5298 | . . . . 5 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
4 | 2, 3 | eqeltri 2886 | . . . 4 ⊢ 𝐴 ∈ V |
5 | 4 | fsn2 6875 | . . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
6 | oveq1 7142 | . . . . . 6 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹( linC ‘𝑍){𝐴}) = ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴})) | |
7 | 6 | adantl 485 | . . . . 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 44756 | . . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
10 | 9 | simpli 487 | . . . . . . 7 ⊢ 𝑍 ∈ LMod |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝑍 ∈ LMod) |
12 | 3z 12003 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
13 | 6nn 11714 | . . . . . . . . . 10 ⊢ 6 ∈ ℕ | |
14 | 13 | nnzi 11994 | . . . . . . . . 9 ⊢ 6 ∈ ℤ |
15 | 8 | zlmodzxzel 44757 | . . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍)) |
16 | 12, 14, 15 | mp2an 691 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍) |
17 | 2, 16 | eqeltri 2886 | . . . . . . 7 ⊢ 𝐴 ∈ (Base‘𝑍) |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝐴 ∈ (Base‘𝑍)) |
19 | simpl 486 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹‘𝐴) ∈ (Base‘ℤring)) | |
20 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
21 | 9 | simpri 489 | . . . . . . 7 ⊢ ℤring = (Scalar‘𝑍) |
22 | eqid 2798 | . . . . . . 7 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
23 | eqid 2798 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑍) = ( ·𝑠 ‘𝑍) | |
24 | 20, 21, 22, 23 | lincvalsng 44825 | . . . . . 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 1368 | . . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
26 | 7, 25 | eqtrd 2833 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
27 | eqid 2798 | . . . . . 6 ⊢ {〈0, 0〉, 〈1, 0〉} = {〈0, 0〉, 〈1, 0〉} | |
28 | eqid 2798 | . . . . . 6 ⊢ (-g‘𝑍) = (-g‘𝑍) | |
29 | zlmodzxzldep.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
30 | 8, 27, 23, 28, 2, 29 | zlmodzxznm 44906 | . . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) |
31 | r19.26 3137 | . . . . . 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 7142 | . . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) | |
33 | 32 | neeq1d 3046 | . . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
34 | 33 | rspcv 3566 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
35 | zringbas 20169 | . . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring) | |
36 | 35 | eqcomi 2807 | . . . . . . . . . . 11 ⊢ (Base‘ℤring) = ℤ |
37 | 36 | eleq2i 2881 | . . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) ↔ (𝐹‘𝐴) ∈ ℤ) |
38 | 37 | biimpi 219 | . . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) → (𝐹‘𝐴) ∈ ℤ) |
39 | 38 | adantr 484 | . . . . . . . 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 484 | . . . . . 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 220 | . . . . 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 3054 | . . 3 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵) |
45 | 5, 44 | sylbi 220 | . 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 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 Vcvv 3441 {csn 4525 {cpr 4527 〈cop 4531 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 0cc0 10526 1c1 10527 2c2 11680 3c3 11681 4c4 11682 6c6 11684 ℤcz 11969 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 -gcsg 18097 LModclmod 19627 ℤringzring 20163 freeLMod cfrlm 20435 linC clinc 44813 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16006 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-0g 16707 df-gsum 16708 df-prds 16713 df-pws 16715 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-lmod 19629 df-lss 19697 df-sra 19937 df-rgmod 19938 df-cnfld 20092 df-zring 20164 df-dsmm 20421 df-frlm 20436 df-linc 44815 |
This theorem is referenced by: ldepsnlinc 44917 |
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