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Theorem 1ewlk 29633

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 14558	. . 3 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → ⟨“𝐼”⟩ ∈ Word dom (iEdg‘𝐺))
2	1	3ad2ant3 1133	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ Word dom (iEdg‘𝐺))
3		ral0 4513	. . . 4 ⊢ ∀𝑘 ∈ ∅ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘))))
4		s1len 14562	. . . . . . . 8 ⊢ (♯‘⟨“𝐼”⟩) = 1
5	4	oveq2i 7424	. . . . . . 7 ⊢ (1..^(♯‘⟨“𝐼”⟩)) = (1..^1)
6		fzo0 13662	. . . . . . 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 3323	. . . 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 257	. . 3 ⊢ (𝐼 ∈ dom (iEdg‘𝐺) → ∀𝑘 ∈ (1..^(♯‘⟨“𝐼”⟩))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘)))))
11	10	3ad2ant3 1133	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ∀𝑘 ∈ (1..^(♯‘⟨“𝐼”⟩))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(⟨“𝐼”⟩‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(⟨“𝐼”⟩‘𝑘)))))
12		eqid 2730	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
13	12	isewlk 29124	. . 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 1163	. 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 709	1 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 Vcvv 3472 ∩ cin 3948 ∅c0 4323 class class class wbr 5149 dom cdm 5677 ‘cfv 6544 (class class class)co 7413 1c1 11115 ≤ cle 11255 − cmin 11450 ℕ₀^*cxnn0 12550 ..^cfzo 13633 ♯chash 14296 Word cword 14470 ⟨“cs1 14551 iEdgciedg 28522 EdgWalks cewlks 29117

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-nn 12219 df-n0 12479 df-z 12565 df-uz 12829 df-fz 13491 df-fzo 13634 df-hash 14297 df-word 14471 df-s1 14552 df-ewlks 29120

This theorem is referenced by: (None)

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