ldepsnlinclem1 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem1 48422

Description: Lemma 1 for ldepsnlinc 48425. (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 8889	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → 𝐹:{𝐵}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.b	. . . . 5 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
3		prex 5437	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
4	2, 3	eqeltri 2837	. . . 4 ⊢ 𝐵 ∈ V
5	4	fsn2 7156	. . 3 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}))
6		oveq1 7438	. . . . . 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 48270	. . . . . . . 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 12649	. . . . . . . . 9 ⊢ 2 ∈ ℤ
13		4z 12651	. . . . . . . . 9 ⊢ 4 ∈ ℤ
14	8	zlmodzxzel 48271	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍))
15	12, 13, 14	mp2an 692	. . . . . . . 8 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍)
16	2, 15	eqeltri 2837	. . . . . . 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 2737	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
20	9	simpri 485	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
21		eqid 2737	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
22		eqid 2737	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
23	19, 20, 21, 22	lincvalsng 48333	. . . . . 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 1373	. . . . 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 2777	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
26		eqid 2737	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
27		eqid 2737	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
28		zlmodzxzldep.a	. . . . . 6 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
29	8, 26, 22, 27, 28, 2	zlmodzxznm 48414	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
30		r19.26 3111	. . . . . 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 7438	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐵) → (𝑖( ·_𝑠 ‘𝑍)𝐵) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
32	31	neeq1d 3000	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐵) → ((𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 ↔ ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
33	32	rspcv 3618	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
34		zringbas 21464	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
35	34	eqcomi 2746	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
36	35	eleq2i 2833	. . . . . . . . . 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 3008	. . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 {csn 4626 {cpr 4628 ⟨cop 4632 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 0cc0 11155 1c1 11156 2c2 12321 3c3 12322 4c4 12323 6c6 12325 ℤcz 12613 Basecbs 17247 Scalarcsca 17300 ·_𝑠 cvsca 17301 -_gcsg 18953 LModclmod 20858 ℤ_ringczring 21457 freeLMod cfrlm 21766 linC clinc 48321

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-dsmm 21752 df-frlm 21767 df-linc 48323

This theorem is referenced by: ldepsnlinc 48425

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