ldepsnlinclem1 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem1 49004

Description: Lemma 1 for ldepsnlinc 49007. (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 8787	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → 𝐹:{𝐵}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.b	. . . . 5 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
3		prex 5368	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
4	2, 3	eqeltri 2835	. . . 4 ⊢ 𝐵 ∈ V
5	4	fsn2 7079	. . 3 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}))
6		oveq1 7364	. . . . . 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 48853	. . . . . . . 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 12551	. . . . . . . . 9 ⊢ 2 ∈ ℤ
13		4z 12553	. . . . . . . . 9 ⊢ 4 ∈ ℤ
14	8	zlmodzxzel 48854	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍))
15	12, 13, 14	mp2an 698	. . . . . . . 8 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍)
16	2, 15	eqeltri 2835	. . . . . . 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 2739	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
20	9	simpri 486	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
21		eqid 2739	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
22		eqid 2739	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
23	19, 20, 21, 22	lincvalsng 48915	. . . . . 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 1379	. . . . 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 2774	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
26		eqid 2739	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
27		eqid 2739	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
28		zlmodzxzldep.a	. . . . . 6 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
29	8, 26, 22, 27, 28, 2	zlmodzxznm 48996	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
30		r19.26 3099	. . . . . 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 7364	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐵) → (𝑖( ·_𝑠 ‘𝑍)𝐵) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
32	31	neeq1d 2993	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐵) → ((𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 ↔ ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
33	32	rspcv 3556	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
34		zringbas 21429	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤ_ring)
35	34	eqcomi 2748	. . . . . . . . . 10 ⊢ (Base‘ℤ_ring) = ℤ
36	35	eleq2i 2831	. . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐵) ∈ ℤ)
37	36	birani 504	. . . . . . . 8 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹‘𝐵) ∈ ℤ)
38	33, 37	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))
39	38	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))
40	30, 39	sylbi 218	. . . . 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))
41	29, 40	ax-mp 5	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
42	25, 41	eqnetrd 3001	. . 3 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
43	5, 42	sylbi 218	. 2 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
44	1, 43	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 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 Vcvv 3431 {csn 4556 {cpr 4558 ⟨cop 4562 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ↑_m cmap 8764 0cc0 11030 1c1 11031 2c2 12228 3c3 12229 4c4 12230 6c6 12232 ℤcz 12516 Basecbs 17171 Scalarcsca 17215 ·_𝑠 cvsca 17216 -_gcsg 18903 LModclmod 20851 ℤ_ringczring 21422 freeLMod cfrlm 21722 linC clinc 48903

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-rp 12935 df-fz 13454 df-fzo 13601 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16214 df-prm 16633 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18904 df-minusg 18905 df-sbg 18906 df-mulg 19036 df-subg 19091 df-cntz 19284 df-cmn 19749 df-abl 19750 df-mgp 20114 df-rng 20126 df-ur 20155 df-ring 20208 df-cring 20209 df-subrng 20519 df-subrg 20543 df-lmod 20853 df-lss 20923 df-sra 21164 df-rgmod 21165 df-cnfld 21349 df-zring 21423 df-dsmm 21708 df-frlm 21723 df-linc 48905

This theorem is referenced by: ldepsnlinc 49007

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