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Theorem ewlkle 27972

Description: An s-walk of edges is also a t-walk of edges if 𝑡 ≤ 𝑠. (Contributed by AV, 4-Jan-2021.)

**Assertion**
Ref	Expression
ewlkle	⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ₀^* ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇))

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

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	ewlkprop 27970	. . 3 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
3		simpl2 1191	. . . . 5 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → 𝐹 ∈ Word dom (iEdg‘𝐺))
4		xnn0xr 12310	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ ℕ₀^* → 𝑇 ∈ ℝ^*)
5	4	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ ℕ₀^* ∧ 𝑇 ∈ ℕ₀^) → 𝑇 ∈ ℝ^)
6		xnn0xr 12310	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ ℕ₀^* → 𝑆 ∈ ℝ^*)
7	6	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ ℕ₀^* ∧ 𝑇 ∈ ℕ₀^) → 𝑆 ∈ ℝ^)
8		fvex 6787	. . . . . . . . . . . . . . . 16 ⊢ ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
9	8	inex1 5241	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V
10		hashxrcl 14072	. . . . . . . . . . . . . . 15 ⊢ ((((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*)
11	9, 10	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ ℕ₀^* ∧ 𝑇 ∈ ℕ₀^) → (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^)
12		xrletr 12892	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ ℝ^* ∧ 𝑆 ∈ ℝ^* ∧ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) ∈ ℝ^*) → ((𝑇 ≤ 𝑆 ∧ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → 𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
13	5, 7, 11, 12	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ ℕ₀^* ∧ 𝑇 ∈ ℕ₀^*) → ((𝑇 ≤ 𝑆 ∧ 𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → 𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
14	13	exp4b 431	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ₀^* → (𝑇 ∈ ℕ₀^* → (𝑇 ≤ 𝑆 → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))))
15	14	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) → (𝑇 ∈ ℕ₀^ → (𝑇 ≤ 𝑆 → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))))
16	15	imp32 419	. . . . . . . . . 10 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → (𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → 𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
17	16	ralimdv 3109	. . . . . . . . 9 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))
18	17	ex 413	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) → ((𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))))
19	18	com23 86	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆) → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))))
20	19	a1d 25	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) → (𝐹 ∈ Word dom (iEdg‘𝐺) → (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))) → ((𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆) → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))))))
21	20	3imp1 1346	. . . . 5 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → ∀𝑘 ∈ (1..^(♯‘𝐹))𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))
22		simpl1l 1223	. . . . . 6 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → 𝐺 ∈ V)
23		simprl 768	. . . . . 6 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → 𝑇 ∈ ℕ₀^*)
24	1	isewlk 27969	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ 𝑇 ∈ ℕ₀^* ∧ 𝐹 ∈ Word dom (iEdg‘𝐺)) → (𝐹 ∈ (𝐺 EdgWalks 𝑇) ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))))
25	22, 23, 3, 24	syl3anc 1370	. . . . 5 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → (𝐹 ∈ (𝐺 EdgWalks 𝑇) ↔ (𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑇 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘)))))))
26	3, 21, 25	mpbir2and 710	. . . 4 ⊢ ((((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) ∧ (𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆)) → 𝐹 ∈ (𝐺 EdgWalks 𝑇))
27	26	ex 413	. . 3 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^) ∧ 𝐹 ∈ Word dom (iEdg‘𝐺) ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘(((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))))) → ((𝑇 ∈ ℕ₀^ ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇)))
28	2, 27	syl 17	. 2 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝑇 ∈ ℕ₀^* ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇)))
29	28	3impib 1115	1 ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ₀^* ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ∩ cin 3886 class class class wbr 5074 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 1c1 10872 ℝ^*cxr 11008 ≤ cle 11010 − cmin 11205 ℕ₀^*cxnn0 12305 ..^cfzo 13382 ♯chash 14044 Word cword 14217 iEdgciedg 27367 EdgWalks cewlks 27962

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-ewlks 27965

This theorem is referenced by: (None)

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