ldepsnlinclem2 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem2 45735

Description: Lemma 2 for ldepsnlinc 45737. (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 8595	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.a	. . . . 5 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
3		prex 5350	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
4	2, 3	eqeltri 2835	. . . 4 ⊢ 𝐴 ∈ V
5	4	fsn2 6990	. . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
6		oveq1 7262	. . . . . 6 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹( linC ‘𝑍){𝐴}) = ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}))
7	6	adantl 481	. . . . 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 45578	. . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤ_ring = (Scalar‘𝑍))
10	9	simpli 483	. . . . . . 7 ⊢ 𝑍 ∈ LMod
11	10	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → 𝑍 ∈ LMod)
12		3z 12283	. . . . . . . . 9 ⊢ 3 ∈ ℤ
13		6nn 11992	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
14	13	nnzi 12274	. . . . . . . . 9 ⊢ 6 ∈ ℤ
15	8	zlmodzxzel 45579	. . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘𝑍))
16	12, 14, 15	mp2an 688	. . . . . . . 8 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘𝑍)
17	2, 16	eqeltri 2835	. . . . . . 7 ⊢ 𝐴 ∈ (Base‘𝑍)
18	17	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → 𝐴 ∈ (Base‘𝑍))
19		simpl 482	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹‘𝐴) ∈ (Base‘ℤ_ring))
20		eqid 2738	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
21	9	simpri 485	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
22		eqid 2738	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
23		eqid 2738	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
24	20, 21, 22, 23	lincvalsng 45645	. . . . . 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 1369	. . . . 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 2778	. . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
27		eqid 2738	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
28		eqid 2738	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
29		zlmodzxzldep.b	. . . . . 6 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
30	8, 27, 23, 28, 2, 29	zlmodzxznm 45726	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
31		r19.26 3094	. . . . . 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 7262	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·_𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
33	32	neeq1d 3002	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
34	33	rspcv 3547	. . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
35		zringbas 20588	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
36	35	eqcomi 2747	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
37	36	eleq2i 2830	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐴) ∈ ℤ)
38	37	biimpi 215	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) → (𝐹‘𝐴) ∈ ℤ)
39	38	adantr 480	. . . . . . . 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 480	. . . . . 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 216	. . . . 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 3010	. . 3 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)
45	5, 44	sylbi 216	. 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 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 Vcvv 3422 {csn 4558 {cpr 4560 ⟨cop 4564 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 0cc0 10802 1c1 10803 2c2 11958 3c3 11959 4c4 11960 6c6 11962 ℤcz 12249 Basecbs 16840 Scalarcsca 16891 ·_𝑠 cvsca 16892 -_gcsg 18494 LModclmod 20038 ℤ_ringzring 20582 freeLMod cfrlm 20863 linC clinc 45633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-prm 16305 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-cntz 18838 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-lmod 20040 df-lss 20109 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-zring 20583 df-dsmm 20849 df-frlm 20864 df-linc 45635

This theorem is referenced by: ldepsnlinc 45737

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