ldepsnlinclem2 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem2 44560

Description: Lemma 2 for ldepsnlinc 44562. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)

**Hypotheses**
Ref	Expression
zlmodzxzldep.z	⊢ 𝑍 = (ℤ_ring freeLMod {0, 1})
zlmodzxzldep.a	⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
zlmodzxzldep.b	⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}

**Assertion**
Ref	Expression
ldepsnlinclem2	⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)

Proof of Theorem ldepsnlinclem2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8427	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.a	. . . . 5 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
3		prex 5332	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
4	2, 3	eqeltri 2909	. . . 4 ⊢ 𝐴 ∈ V
5	4	fsn2 6897	. . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
6		oveq1 7162	. . . . . 6 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹( linC ‘𝑍){𝐴}) = ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}))
7	6	adantl 484	. . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) = ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}))
8		zlmodzxzldep.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_ring freeLMod {0, 1})
9	8	zlmodzxzlmod 44401	. . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤ_ring = (Scalar‘𝑍))
10	9	simpli 486	. . . . . . 7 ⊢ 𝑍 ∈ LMod
11	10	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → 𝑍 ∈ LMod)
12		3z 12014	. . . . . . . . 9 ⊢ 3 ∈ ℤ
13		6nn 11725	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
14	13	nnzi 12005	. . . . . . . . 9 ⊢ 6 ∈ ℤ
15	8	zlmodzxzel 44402	. . . . . . . . 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 2909	. . . . . . 7 ⊢ 𝐴 ∈ (Base‘𝑍)
18	17	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → 𝐴 ∈ (Base‘𝑍))
19		simpl 485	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹‘𝐴) ∈ (Base‘ℤ_ring))
20		eqid 2821	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
21	9	simpri 488	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
22		eqid 2821	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
23		eqid 2821	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
24	20, 21, 22, 23	lincvalsng 44470	. . . . . 6 ⊢ ((𝑍 ∈ LMod ∧ 𝐴 ∈ (Base‘𝑍) ∧ (𝐹‘𝐴) ∈ (Base‘ℤ_ring)) → ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
25	11, 18, 19, 24	syl3anc 1367	. . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
26	7, 25	eqtrd 2856	. . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
27		eqid 2821	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
28		eqid 2821	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
29		zlmodzxzldep.b	. . . . . 6 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
30	8, 27, 23, 28, 2, 29	zlmodzxznm 44551	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
31		r19.26 3170	. . . . . 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 7162	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·_𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
33	32	neeq1d 3075	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
34	33	rspcv 3617	. . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
35		zringbas 20622	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
36	35	eqcomi 2830	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
37	36	eleq2i 2904	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐴) ∈ ℤ)
38	37	biimpi 218	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) → (𝐹‘𝐴) ∈ ℤ)
39	38	adantr 483	. . . . . . . 8 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹‘𝐴) ∈ ℤ)
40	34, 39	syl11 33	. . . . . . 7 ⊢ (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 → (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
41	40	adantr 483	. . . . . 6 ⊢ ((∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ ∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
42	31, 41	sylbi 219	. . . . 5 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
43	30, 42	ax-mp 5	. . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵)
44	26, 43	eqnetrd 3083	. . 3 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)
45	5, 44	sylbi 219	. 2 ⊢ (𝐹:{𝐴}⟶(Base‘ℤ_ring) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)
46	1, 45	syl 17	1 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 Vcvv 3494 {csn 4566 {cpr 4568 ⟨cop 4572 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑_m cmap 8405 0cc0 10536 1c1 10537 2c2 11691 3c3 11692 4c4 11693 6c6 11695 ℤcz 11980 Basecbs 16482 Scalarcsca 16567 ·_𝑠 cvsca 16568 -_gcsg 18104 LModclmod 19633 ℤ_ringzring 20616 freeLMod cfrlm 20889 linC clinc 44458

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-sup 8905 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-dvds 15607 df-prm 16015 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-0g 16714 df-gsum 16715 df-prds 16720 df-pws 16722 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-cntz 18446 df-cmn 18907 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-subrg 19532 df-lmod 19635 df-lss 19703 df-sra 19943 df-rgmod 19944 df-cnfld 20545 df-zring 20617 df-dsmm 20875 df-frlm 20890 df-linc 44460

This theorem is referenced by: ldepsnlinc 44562

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