ldepsnlinclem1 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem1 48234

Description: Lemma 1 for ldepsnlinc 48237. (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 8907	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → 𝐹:{𝐵}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.b	. . . . 5 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
3		prex 5452	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
4	2, 3	eqeltri 2840	. . . 4 ⊢ 𝐵 ∈ V
5	4	fsn2 7170	. . 3 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}))
6		oveq1 7455	. . . . . 6 ⊢ (𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩} → (𝐹( linC ‘𝑍){𝐵}) = ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}))
7	6	adantl 481	. . . . 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 48079	. . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤ_ring = (Scalar‘𝑍))
10	9	simpli 483	. . . . . . 7 ⊢ 𝑍 ∈ LMod
11	10	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → 𝑍 ∈ LMod)
12		2z 12675	. . . . . . . . 9 ⊢ 2 ∈ ℤ
13		4z 12677	. . . . . . . . 9 ⊢ 4 ∈ ℤ
14	8	zlmodzxzel 48080	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍))
15	12, 13, 14	mp2an 691	. . . . . . . 8 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍)
16	2, 15	eqeltri 2840	. . . . . . 7 ⊢ 𝐵 ∈ (Base‘𝑍)
17	16	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → 𝐵 ∈ (Base‘𝑍))
18		simpl 482	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹‘𝐵) ∈ (Base‘ℤ_ring))
19		eqid 2740	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
20	9	simpri 485	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
21		eqid 2740	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
22		eqid 2740	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
23	19, 20, 21, 22	lincvalsng 48145	. . . . . 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 2780	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
26		eqid 2740	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
27		eqid 2740	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
28		zlmodzxzldep.a	. . . . . 6 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
29	8, 26, 22, 27, 28, 2	zlmodzxznm 48226	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
30		r19.26 3117	. . . . . 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 7455	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐵) → (𝑖( ·_𝑠 ‘𝑍)𝐵) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
32	31	neeq1d 3006	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐵) → ((𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 ↔ ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
33	32	rspcv 3631	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
34		zringbas 21487	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
35	34	eqcomi 2749	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
36	35	eleq2i 2836	. . . . . . . . . 10 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐵) ∈ ℤ)
37	36	biimpi 216	. . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) → (𝐹‘𝐵) ∈ ℤ)
38	37	adantr 480	. . . . . . . 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 481	. . . . . 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 217	. . . . 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 3014	. . 3 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
44	5, 43	sylbi 217	. 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 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 Vcvv 3488 {csn 4648 {cpr 4650 ⟨cop 4654 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 0cc0 11184 1c1 11185 2c2 12348 3c3 12349 4c4 12350 6c6 12352 ℤcz 12639 Basecbs 17258 Scalarcsca 17314 ·_𝑠 cvsca 17315 -_gcsg 18975 LModclmod 20880 ℤ_ringczring 21480 freeLMod cfrlm 21789 linC clinc 48133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-prm 16719 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-lmod 20882 df-lss 20953 df-sra 21195 df-rgmod 21196 df-cnfld 21388 df-zring 21481 df-dsmm 21775 df-frlm 21790 df-linc 48135

This theorem is referenced by: ldepsnlinc 48237

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