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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ldepsnlinclem2 | Structured version Visualization version GIF version | ||
| Description: Lemma 2 for ldepsnlinc 48425. (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 8889 | . 2 ⊢ (𝐹 ∈ ((Base‘ℤring) ↑m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤring)) | |
| 2 | zlmodzxzldep.a | . . . . 5 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
| 3 | prex 5437 | . . . . 5 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
| 4 | 2, 3 | eqeltri 2837 | . . . 4 ⊢ 𝐴 ∈ V |
| 5 | 4 | fsn2 7156 | . . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
| 6 | oveq1 7438 | . . . . . 6 ⊢ (𝐹 = {〈𝐴, (𝐹‘𝐴)〉} → (𝐹( linC ‘𝑍){𝐴}) = ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴})) | |
| 7 | 6 | adantl 481 | . . . . 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 48270 | . . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍)) |
| 10 | 9 | simpli 483 | . . . . . . 7 ⊢ 𝑍 ∈ LMod |
| 11 | 10 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝑍 ∈ LMod) |
| 12 | 3z 12650 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
| 13 | 6nn 12355 | . . . . . . . . . 10 ⊢ 6 ∈ ℕ | |
| 14 | 13 | nnzi 12641 | . . . . . . . . 9 ⊢ 6 ∈ ℤ |
| 15 | 8 | zlmodzxzel 48271 | . . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍)) |
| 16 | 12, 14, 15 | mp2an 692 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ (Base‘𝑍) |
| 17 | 2, 16 | eqeltri 2837 | . . . . . . 7 ⊢ 𝐴 ∈ (Base‘𝑍) |
| 18 | 17 | a1i 11 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → 𝐴 ∈ (Base‘𝑍)) |
| 19 | simpl 482 | . . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹‘𝐴) ∈ (Base‘ℤring)) | |
| 20 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
| 21 | 9 | simpri 485 | . . . . . . 7 ⊢ ℤring = (Scalar‘𝑍) |
| 22 | eqid 2737 | . . . . . . 7 ⊢ (Base‘ℤring) = (Base‘ℤring) | |
| 23 | eqid 2737 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑍) = ( ·𝑠 ‘𝑍) | |
| 24 | 20, 21, 22, 23 | lincvalsng 48333 | . . . . . 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 1373 | . . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → ({〈𝐴, (𝐹‘𝐴)〉} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
| 26 | 7, 25 | eqtrd 2777 | . . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) |
| 27 | eqid 2737 | . . . . . 6 ⊢ {〈0, 0〉, 〈1, 0〉} = {〈0, 0〉, 〈1, 0〉} | |
| 28 | eqid 2737 | . . . . . 6 ⊢ (-g‘𝑍) = (-g‘𝑍) | |
| 29 | zlmodzxzldep.b | . . . . . 6 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
| 30 | 8, 27, 23, 28, 2, 29 | zlmodzxznm 48414 | . . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·𝑠 ‘𝑍)𝐵) ≠ 𝐴) |
| 31 | r19.26 3111 | . . . . . 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 7438 | . . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴)) | |
| 33 | 32 | neeq1d 3000 | . . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
| 34 | 33 | rspcv 3618 | . . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·𝑠 ‘𝑍)𝐴) ≠ 𝐵)) |
| 35 | zringbas 21464 | . . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤring) | |
| 36 | 35 | eqcomi 2746 | . . . . . . . . . . 11 ⊢ (Base‘ℤring) = ℤ |
| 37 | 36 | eleq2i 2833 | . . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) ↔ (𝐹‘𝐴) ∈ ℤ) |
| 38 | 37 | biimpi 216 | . . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤring) → (𝐹‘𝐴) ∈ ℤ) |
| 39 | 38 | adantr 480 | . . . . . . . 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 480 | . . . . . 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 217 | . . . . 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 3008 | . . 3 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤring) ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵) |
| 45 | 5, 44 | sylbi 217 | . 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 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 {csn 4626 {cpr 4628 〈cop 4632 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 1c1 11156 2c2 12321 3c3 12322 4c4 12323 6c6 12325 ℤcz 12613 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 -gcsg 18953 LModclmod 20858 ℤringczring 21457 freeLMod cfrlm 21766 linC clinc 48321 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-dsmm 21752 df-frlm 21767 df-linc 48323 |
| This theorem is referenced by: ldepsnlinc 48425 |
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