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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldepsnlinclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for ldepsnlinc 44491. (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 |
---|---|
ldepsnlinclem1 | ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8417 | . 2 ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐵}) → 𝐹:{𝐵}⟶(Base‘ℤring)) | |
2 | zlmodzxzldep.b | . . . . 5 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
3 | prex 5323 | . . . . 5 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
4 | 2, 3 | eqeltri 2906 | . . . 4 ⊢ 𝐵 ∈ V |
5 | 4 | fsn2 6890 | . . 3 ⊢ (𝐹:{𝐵}⟶(Base‘ℤring) ↔ ((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉})) |
6 | oveq1 7152 | . . . . . 6 ⊢ (𝐹 = {〈𝐵, (𝐹‘𝐵)〉} → (𝐹( linC ‘𝑍){𝐵}) = ({〈𝐵, (𝐹‘𝐵)〉} ( linC ‘𝑍){𝐵})) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → (𝐹( linC ‘𝑍){𝐵}) = ({〈𝐵, (𝐹‘𝐵)〉} ( linC ‘𝑍){𝐵})) |
8 | zlmodzxzldep.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
9 | 8 | zlmodzxzlmod 44330 | . . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
10 | 9 | simpli 484 | . . . . . . 7 ⊢ 𝑍 ∈ LMod |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → 𝑍 ∈ LMod) |
12 | 2z 12002 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
13 | 4z 12004 | . . . . . . . . 9 ⊢ 4 ∈ ℤ | |
14 | 8 | zlmodzxzel 44331 | . . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {〈0, 2〉, 〈1, 4〉} ∈ (Base‘𝑍)) |
15 | 12, 13, 14 | mp2an 688 | . . . . . . . 8 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ (Base‘𝑍) |
16 | 2, 15 | eqeltri 2906 | . . . . . . 7 ⊢ 𝐵 ∈ (Base‘𝑍) |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → 𝐵 ∈ (Base‘𝑍)) |
18 | simpl 483 | . . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → (𝐹‘𝐵) ∈ (Base‘ℤring)) | |
19 | eqid 2818 | . . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
20 | 9 | simpri 486 | . . . . . . 7 ⊢ ℤring = (Scalar‘𝑍) |
21 | eqid 2818 | . . . . . . 7 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
22 | eqid 2818 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑍) = ( ·𝑠 ‘𝑍) | |
23 | 19, 20, 21, 22 | lincvalsng 44399 | . . . . . 6 ⊢ ((𝑍 ∈ LMod ∧ 𝐵 ∈ (Base‘𝑍) ∧ (𝐹‘𝐵) ∈ (Base‘ℤring)) → ({〈𝐵, (𝐹‘𝐵)〉} ( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵)) |
24 | 11, 17, 18, 23 | syl3anc 1363 | . . . . 5 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → ({〈𝐵, (𝐹‘𝐵)〉} ( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵)) |
25 | 7, 24 | eqtrd 2853 | . . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → (𝐹( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵)) |
26 | eqid 2818 | . . . . . 6 ⊢ {〈0, 0〉, 〈1, 0〉} = {〈0, 0〉, 〈1, 0〉} | |
27 | eqid 2818 | . . . . . 6 ⊢ (-g‘𝑍) = (-g‘𝑍) | |
28 | zlmodzxzldep.a | . . . . . 6 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
29 | 8, 26, 22, 27, 28, 2 | zlmodzxznm 44480 | . . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) |
30 | r19.26 3167 | . . . . . 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)) | |
31 | oveq1 7152 | . . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐵) → (𝑖( ·𝑠 ‘𝑍)𝐵) = ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵)) | |
32 | 31 | neeq1d 3072 | . . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐵) → ((𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴 ↔ ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴)) |
33 | 32 | rspcv 3615 | . . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴 → ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴)) |
34 | zringbas 20551 | . . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring) | |
35 | 34 | eqcomi 2827 | . . . . . . . . . . 11 ⊢ (Base‘ℤring) = ℤ |
36 | 35 | eleq2i 2901 | . . . . . . . . . 10 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤring) ↔ (𝐹‘𝐵) ∈ ℤ) |
37 | 36 | biimpi 217 | . . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤring) → (𝐹‘𝐵) ∈ ℤ) |
38 | 37 | adantr 481 | . . . . . . . 8 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → (𝐹‘𝐵) ∈ ℤ) |
39 | 33, 38 | syl11 33 | . . . . . . 7 ⊢ (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴 → (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴)) |
40 | 39 | adantl 482 | . . . . . 6 ⊢ ((∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ ∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴)) |
41 | 30, 40 | sylbi 218 | . . . . 5 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴)) |
42 | 29, 41 | ax-mp 5 | . . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → ((𝐹‘𝐵)( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) |
43 | 25, 42 | eqnetrd 3080 | . . 3 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐵, (𝐹‘𝐵)〉}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴) |
44 | 5, 43 | sylbi 218 | . 2 ⊢ (𝐹:{𝐵}⟶(Base‘ℤring) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴) |
45 | 1, 44 | syl 17 | 1 ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 {csn 4557 {cpr 4559 〈cop 4563 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 0cc0 10525 1c1 10526 2c2 11680 3c3 11681 4c4 11682 6c6 11684 ℤcz 11969 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 -gcsg 18043 LModclmod 19563 ℤringzring 20545 freeLMod cfrlm 20818 linC clinc 44387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 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 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-prm 16004 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-0g 16703 df-gsum 16704 df-prds 16709 df-pws 16711 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-cntz 18385 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-subrg 19462 df-lmod 19565 df-lss 19633 df-sra 19873 df-rgmod 19874 df-cnfld 20474 df-zring 20546 df-dsmm 20804 df-frlm 20819 df-linc 44389 |
This theorem is referenced by: ldepsnlinc 44491 |
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