ldepsnlinclem1 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem1 46518

Description: Lemma 1 for ldepsnlinc 46521. (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
ldepsnlinclem1	⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)

Proof of Theorem ldepsnlinclem1 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8783	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → 𝐹:{𝐵}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.b	. . . . 5 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
3		prex 5387	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
4	2, 3	eqeltri 2834	. . . 4 ⊢ 𝐵 ∈ V
5	4	fsn2 7078	. . 3 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}))
6		oveq1 7360	. . . . . 6 ⊢ (𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩} → (𝐹( linC ‘𝑍){𝐵}) = ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}))
7	6	adantl 482	. . . . 5 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) = ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}))
8		zlmodzxzldep.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_ring freeLMod {0, 1})
9	8	zlmodzxzlmod 46362	. . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤ_ring = (Scalar‘𝑍))
10	9	simpli 484	. . . . . . 7 ⊢ 𝑍 ∈ LMod
11	10	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → 𝑍 ∈ LMod)
12		2z 12531	. . . . . . . . 9 ⊢ 2 ∈ ℤ
13		4z 12533	. . . . . . . . 9 ⊢ 4 ∈ ℤ
14	8	zlmodzxzel 46363	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍))
15	12, 13, 14	mp2an 690	. . . . . . . 8 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍)
16	2, 15	eqeltri 2834	. . . . . . 7 ⊢ 𝐵 ∈ (Base‘𝑍)
17	16	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → 𝐵 ∈ (Base‘𝑍))
18		simpl 483	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹‘𝐵) ∈ (Base‘ℤ_ring))
19		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
20	9	simpri 486	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
21		eqid 2736	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
22		eqid 2736	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
23	19, 20, 21, 22	lincvalsng 46429	. . . . . 6 ⊢ ((𝑍 ∈ LMod ∧ 𝐵 ∈ (Base‘𝑍) ∧ (𝐹‘𝐵) ∈ (Base‘ℤ_ring)) → ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
24	11, 17, 18, 23	syl3anc 1371	. . . . 5 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
25	7, 24	eqtrd 2776	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
26		eqid 2736	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
27		eqid 2736	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
28		zlmodzxzldep.a	. . . . . 6 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
29	8, 26, 22, 27, 28, 2	zlmodzxznm 46510	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
30		r19.26 3112	. . . . . 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 7360	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐵) → (𝑖( ·_𝑠 ‘𝑍)𝐵) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
32	31	neeq1d 3001	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐵) → ((𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 ↔ ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
33	32	rspcv 3575	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
34		zringbas 20860	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
35	34	eqcomi 2745	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
36	35	eleq2i 2829	. . . . . . . . . 10 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐵) ∈ ℤ)
37	36	biimpi 215	. . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) → (𝐹‘𝐵) ∈ ℤ)
38	37	adantr 481	. . . . . . . 8 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹‘𝐵) ∈ ℤ)
39	33, 38	syl11 33	. . . . . . 7 ⊢ (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 → (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 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‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
41	30, 40	sylbi 216	. . . . 5 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
42	29, 41	ax-mp 5	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
43	25, 42	eqnetrd 3009	. . 3 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
44	5, 43	sylbi 216	. 2 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
45	1, 44	syl 17	1 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 Vcvv 3443 {csn 4584 {cpr 4586 ⟨cop 4590 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 ↑_m cmap 8761 0cc0 11047 1c1 11048 2c2 12204 3c3 12205 4c4 12206 6c6 12208 ℤcz 12495 Basecbs 17075 Scalarcsca 17128 ·_𝑠 cvsca 17129 -_gcsg 18742 LModclmod 20307 ℤ_ringczring 20854 freeLMod cfrlm 21137 linC clinc 46417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 ax-addf 11126 ax-mulf 11127

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-2o 8409 df-er 8644 df-map 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-sup 9374 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-rp 12908 df-fz 13417 df-fzo 13560 df-seq 13899 df-exp 13960 df-hash 14223 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-dvds 16129 df-prm 16540 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-starv 17140 df-sca 17141 df-vsca 17142 df-ip 17143 df-tset 17144 df-ple 17145 df-ds 17147 df-unif 17148 df-hom 17149 df-cco 17150 df-0g 17315 df-gsum 17316 df-prds 17321 df-pws 17323 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-grp 18743 df-minusg 18744 df-sbg 18745 df-mulg 18864 df-subg 18916 df-cntz 19088 df-cmn 19555 df-mgp 19888 df-ur 19905 df-ring 19952 df-cring 19953 df-subrg 20205 df-lmod 20309 df-lss 20378 df-sra 20618 df-rgmod 20619 df-cnfld 20782 df-zring 20855 df-dsmm 21123 df-frlm 21138 df-linc 46419

This theorem is referenced by: ldepsnlinc 46521

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