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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldepsnlinclem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for ldepsnlinc 47762. (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 8868 | . 2 ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤring)) | |
2 | zlmodzxzldep.a | . . . . 5 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
3 | prex 5434 | . . . . 5 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
4 | 2, 3 | eqeltri 2821 | . . . 4 ⊢ 𝐴 ∈ V |
5 | 4 | fsn2 7145 | . . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
6 | oveq1 7426 | . . . . . 6 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹( linC ‘𝑍){𝐴}) = ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴})) | |
7 | 6 | adantl 480 | . . . . 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 47604 | . . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
10 | 9 | simpli 482 | . . . . . . 7 ⊢ 𝑍 ∈ LMod |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝑍 ∈ LMod) |
12 | 3z 12628 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
13 | 6nn 12334 | . . . . . . . . . 10 ⊢ 6 ∈ ℕ | |
14 | 13 | nnzi 12619 | . . . . . . . . 9 ⊢ 6 ∈ ℤ |
15 | 8 | zlmodzxzel 47605 | . . . . . . . . 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 2821 | . . . . . . 7 ⊢ 𝐴 ∈ (Base‘𝑍) |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝐴 ∈ (Base‘𝑍)) |
19 | simpl 481 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹‘𝐴) ∈ (Base‘ℤring)) | |
20 | eqid 2725 | . . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
21 | 9 | simpri 484 | . . . . . . 7 ⊢ ℤring = (Scalar‘𝑍) |
22 | eqid 2725 | . . . . . . 7 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
23 | eqid 2725 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑍) = ( ·𝑠 ‘𝑍) | |
24 | 20, 21, 22, 23 | lincvalsng 47670 | . . . . . 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 2765 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
27 | eqid 2725 | . . . . . 6 ⊢ {〈0, 0〉, 〈1, 0〉} = {〈0, 0〉, 〈1, 0〉} | |
28 | eqid 2725 | . . . . . 6 ⊢ (-g‘𝑍) = (-g‘𝑍) | |
29 | zlmodzxzldep.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
30 | 8, 27, 23, 28, 2, 29 | zlmodzxznm 47751 | . . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) |
31 | r19.26 3100 | . . . . . 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 7426 | . . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) | |
33 | 32 | neeq1d 2989 | . . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
34 | 33 | rspcv 3602 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
35 | zringbas 21396 | . . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring) | |
36 | 35 | eqcomi 2734 | . . . . . . . . . . 11 ⊢ (Base‘ℤring) = ℤ |
37 | 36 | eleq2i 2817 | . . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) ↔ (𝐹‘𝐴) ∈ ℤ) |
38 | 37 | biimpi 215 | . . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) → (𝐹‘𝐴) ∈ ℤ) |
39 | 38 | adantr 479 | . . . . . . . 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 479 | . . . . . 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 2997 | . . 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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 Vcvv 3461 {csn 4630 {cpr 4632 〈cop 4636 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 0cc0 11140 1c1 11141 2c2 12300 3c3 12301 4c4 12302 6c6 12304 ℤcz 12591 Basecbs 17183 Scalarcsca 17239 ·𝑠 cvsca 17240 -gcsg 18900 LModclmod 20755 ℤringczring 21389 freeLMod cfrlm 21697 linC clinc 47658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-sup 9467 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-prm 16646 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-0g 17426 df-gsum 17427 df-prds 17432 df-pws 17434 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19032 df-subg 19086 df-cntz 19280 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-subrng 20495 df-subrg 20520 df-lmod 20757 df-lss 20828 df-sra 21070 df-rgmod 21071 df-cnfld 21297 df-zring 21390 df-dsmm 21683 df-frlm 21698 df-linc 47660 |
This theorem is referenced by: ldepsnlinc 47762 |
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