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Theorem 1ewlk 30096

Description: A sequence of 1 edge is an s-walk of edges for all s. (Contributed by AV, 5-Jan-2021.)

**Assertion**
Ref	Expression
1ewlk	⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆))

Proof of Theorem 1ewlk Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		s1cl 14620	. . 3 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → ⟨“𝐼”⟩ ∈ Word dom (iEdg‘𝐺))
2	1	3ad2ant3 1135	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ Word dom (iEdg‘𝐺))
3		ral0 4488	. . . 4 ⊢ ∀𝑘 ∈ ∅ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘))))
4		s1len 14624	. . . . . . . 8 ⊢ (♯‘⟨“𝐼”⟩) = 1
5	4	oveq2i 7416	. . . . . . 7 ⊢ (1..^(♯‘⟨“𝐼”⟩)) = (1..^1)
6		fzo0 13700	. . . . . . 7 ⊢ (1..^1) = ∅
7	5, 6	eqtri 2758	. . . . . 6 ⊢ (1..^(♯‘⟨“𝐼”⟩)) = ∅
8	7	a1i 11	. . . . 5 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → (1..^(♯‘⟨“𝐼”⟩)) = ∅)
9	8	raleqdv 3305	. . . 4 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(♯‘⟨“𝐼”⟩))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘)))) ↔ ∀𝑘 ∈ ∅ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘))))))
10	3, 9	mpbiri 258	. . 3 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → ∀𝑘 ∈ (1..^(♯‘⟨“𝐼”⟩))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘)))))
11	10	3ad2ant3 1135	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ∀𝑘 ∈ (1..^(♯‘⟨“𝐼”⟩))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘)))))
12		eqid 2735	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
13	12	isewlk 29582	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ ⟨“𝐼”⟩ ∈ Word dom (iEdg‘𝐺)) → (⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆) ↔ (⟨“𝐼”⟩ ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘⟨“𝐼”⟩))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘)))))))
14	1, 13	syl3an3 1165	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → (⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆) ↔ (⟨“𝐼”⟩ ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘⟨“𝐼”⟩))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘)))))))
15	2, 11, 14	mpbir2and 713	1 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 Vcvv 3459 ∩ cin 3925 ∅c0 4308 class class class wbr 5119 dom cdm 5654 ‘cfv 6531 (class class class)co 7405 1c1 11130 ≤ cle 11270 − cmin 11466 ℕ₀^*cxnn0 12574 ..^cfzo 13671 ♯chash 14348 Word cword 14531 ⟨“cs1 14613 iEdgciedg 28976 EdgWalks cewlks 29575

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-s1 14614 df-ewlks 29578

This theorem is referenced by: (None)

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