ldepsnlinclem2 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem2 48495

Description: Lemma 2 for ldepsnlinc 48497. (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 8822	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.a	. . . . 5 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
3		prex 5392	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
4	2, 3	eqeltri 2824	. . . 4 ⊢ 𝐴 ∈ V
5	4	fsn2 7108	. . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
6		oveq1 7394	. . . . . 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 48342	. . . . . . . 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 12566	. . . . . . . . 9 ⊢ 3 ∈ ℤ
13		6nn 12275	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
14	13	nnzi 12557	. . . . . . . . 9 ⊢ 6 ∈ ℤ
15	8	zlmodzxzel 48343	. . . . . . . . 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 2824	. . . . . . 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 2729	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
21	9	simpri 485	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
22		eqid 2729	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
23		eqid 2729	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
24	20, 21, 22, 23	lincvalsng 48405	. . . . . 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 1373	. . . . 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 2764	. . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
27		eqid 2729	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
28		eqid 2729	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
29		zlmodzxzldep.b	. . . . . 6 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
30	8, 27, 23, 28, 2, 29	zlmodzxznm 48486	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
31		r19.26 3091	. . . . . 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 7394	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·_𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
33	32	neeq1d 2984	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
34	33	rspcv 3584	. . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
35		zringbas 21363	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
36	35	eqcomi 2738	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
37	36	eleq2i 2820	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐴) ∈ ℤ)
38	37	biimpi 216	. . . . . . . . 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 217	. . . . 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 2992	. . 3 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)
45	5, 44	sylbi 217	. 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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3447 {csn 4589 {cpr 4591 ⟨cop 4595 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ↑_m cmap 8799 0cc0 11068 1c1 11069 2c2 12241 3c3 12242 4c4 12243 6c6 12245 ℤcz 12529 Basecbs 17179 Scalarcsca 17223 ·_𝑠 cvsca 17224 -_gcsg 18867 LModclmod 20766 ℤ_ringczring 21356 freeLMod cfrlm 21655 linC clinc 48393

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-rp 12952 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-prm 16642 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-cntz 19249 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-subrng 20455 df-subrg 20479 df-lmod 20768 df-lss 20838 df-sra 21080 df-rgmod 21081 df-cnfld 21265 df-zring 21357 df-dsmm 21641 df-frlm 21656 df-linc 48395

This theorem is referenced by: ldepsnlinc 48497

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