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Theorem ewlkprop 26555

Description: Properties of an s-walk of edges. (Contributed by AV, 4-Jan-2021.)

**Hypothesis**
Ref	Expression
ewlksfval.i	⊢ 𝐼 = (iEdg‘𝐺)

**Assertion**
Ref	Expression
ewlkprop	⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))

Distinct variable groups: 𝑘,𝐺 𝑆,𝑘 𝑘,𝐹 Allowed substitution hint: 𝐼(𝑘)

Proof of Theorem ewlkprop Dummy variables 𝑓 𝑔 𝑖 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ewlks 26550	. . 3 ⊢ EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ₀^* ↦ {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))})
2	1	elmpt2cl 6918	. 2 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*))
3		simpr 476	. . 3 ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^)) → (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^))
4		ewlksfval.i	. . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺)
5	4	isewlk 26554	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆)) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) ↔ (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))))
6	5	3expa 1284	. . . . . . 7 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆)) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) ↔ (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))))
7	6	biimpd 219	. . . . . 6 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆)) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))))
8	7	expcom 450	. . . . 5 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))))
9	8	pm2.43a 54	. . . 4 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) → (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))))
10	9	imp 444	. . 3 ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*)) → (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))
11		3anass 1059	. . 3 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))) ↔ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))))
12	3, 10, 11	sylanbrc 699	. 2 ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^)) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))
13	2, 12	mpdan 703	1 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 {cab 2637 ∀wral 2941 Vcvv 3231 [wsbc 3468 ∩ cin 3606 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 (class class class)co 6690 1c1 9975 ≤ cle 10113 − cmin 10304 ℕ₀^*cxnn0 11401 ..^cfzo 12504 #chash 13157 Word cword 13323 iEdgciedg 25920 EdgWalks cewlks 26547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-er 7787 df-map 7901 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 df-ewlks 26550

This theorem is referenced by: ewlkinedg 26556 ewlkle 26557 upgrewlkle2 26558

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