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Theorem ewlkprop 29862

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 29857	. . 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	elmpocl 7641	. 2 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*))
3		simpr 489	. . 3 ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^)) → (𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^))
4		ewlksfval.i	. . . . . . . . 9 ⊢ 𝐼 = (iEdg‘𝐺)
5	4	isewlk 29861	. . . . . . . 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 1134	. . . . . . 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 232	. . . . . 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 418	. . . . 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 55	. . . 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 411	. . 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 1109	. . 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 594	. 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 699	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 {cab 2743 ∀wral 3079 Vcvv 3457 [wsbc 3747 ∩ cin 3906 class class class wbr 5105 dom cdm 5652 ‘cfv 6525 (class class class)co 7400 1c1 11089 ≤ cle 11232 − cmin 11429 ℕ₀^*cxnn0 12568 ..^cfzo 13673 ♯chash 14357 Word cword 14540 iEdgciedg 29256 EdgWalks cewlks 29854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-fzo 13674 df-hash 14358 df-word 14541 df-ewlks 29857

This theorem is referenced by: ewlkinedg 29863 ewlkle 29864 upgrewlkle2 29865

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