clwlksf1clwwlklem2OLD - Metamath Proof Explorer

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Theorem clwlksf1clwwlklem2OLD 27406

Description: Obsolete as of 24-May-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
clwlksfclwwlkOLD.1	⊢ 𝐴 = (1^st ‘𝑐)
clwlksfclwwlkOLD.2	⊢ 𝐵 = (2^nd ‘𝑐)
clwlksfclwwlkOLD.c	⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlkOLD.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))

**Assertion**
Ref	Expression
clwlksf1clwwlklem2OLD	⊢ (𝑊 ∈ 𝐶 → ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘𝑁))

Distinct variable groups: 𝐺,𝑐 𝑁,𝑐 𝑊,𝑐 𝐶,𝑐 𝐹,𝑐 Allowed substitution hints: 𝐴(𝑐) 𝐵(𝑐)

**Proof of Theorem clwlksf1clwwlklem2OLD**
Step	Hyp	Ref	Expression
1		clwlksfclwwlkOLD.1	. . 3 ⊢ 𝐴 = (1^st ‘𝑐)
2		clwlksfclwwlkOLD.2	. . 3 ⊢ 𝐵 = (2^nd ‘𝑐)
3		clwlksfclwwlkOLD.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4		clwlksfclwwlkOLD.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
5	1, 2, 3, 4	clwlksf1clwwlklem0OLD 27404	. 2 ⊢ (𝑊 ∈ 𝐶 → (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))) ∧ (♯‘(1^st ‘𝑊)) = 𝑁))
6		fveq2 6412	. . . . . 6 ⊢ ((♯‘(1^st ‘𝑊)) = 𝑁 → ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊))) = ((2^nd ‘𝑊)‘𝑁))
7	6	eqeq2d 2810	. . . . 5 ⊢ ((♯‘(1^st ‘𝑊)) = 𝑁 → (((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊))) ↔ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘𝑁)))
8	7	biimpcd 241	. . . 4 ⊢ (((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊))) → ((♯‘(1^st ‘𝑊)) = 𝑁 → ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘𝑁)))
9	8	3ad2ant3 1166	. . 3 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))) → ((♯‘(1^st ‘𝑊)) = 𝑁 → ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘𝑁)))
10	9	imp 396	. 2 ⊢ ((((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘𝑁))
11	5, 10	syl 17	1 ⊢ (𝑊 ∈ 𝐶 → ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 {crab 3094 ⟨cop 4375 ↦ cmpt 4923 dom cdm 5313 ⟶wf 6098 ‘cfv 6102 (class class class)co 6879 1^st c1st 7400 2^nd c2nd 7401 0cc0 10225 ...cfz 12579 ♯chash 13369 Word cword 13533 substr csubstr 13663 Vtxcvtx 26230 iEdgciedg 26231 ClWalkscclwlks 27023

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-ifp 1087 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-er 7983 df-map 8098 df-pm 8099 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-card 9052 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-fzo 12720 df-hash 13370 df-word 13534 df-wlks 26848 df-clwlks 27024

This theorem is referenced by: clwlksf1clwwlklemOLD 27408

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