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Theorem ewlkle 29692

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 2739	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	ewlkprop 29690	. . 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 1199	. . . . 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 12506	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ ℕ₀^* → 𝑇 ∈ ℝ^*)
5	4	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ ℕ₀^* ∧ 𝑇 ∈ ℕ₀^) → 𝑇 ∈ ℝ^)
6		xnn0xr 12506	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ ℕ₀^* → 𝑆 ∈ ℝ^*)
7	6	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ ℕ₀^* ∧ 𝑇 ∈ ℕ₀^) → 𝑆 ∈ ℝ^)
8		fvex 6840	. . . . . . . . . . . . . . . 16 ⊢ ((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∈ V
9	8	inex1 5245	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘(𝐹‘(𝑘 − 1))) ∩ ((iEdg‘𝐺)‘(𝐹‘𝑘))) ∈ V
10		hashxrcl 14310	. . . . . . . . . . . . . . 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 13100	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 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 3153	. . . . . . . . 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 1354	. . . . 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 1231	. . . . . 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 776	. . . . . 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 29689	. . . . . 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 1379	. . . . 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 719	. . . 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 1122	1 ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ₀^* ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 ∈ wcel 2119 ∀wral 3053 Vcvv 3431 ∩ cin 3882 class class class wbr 5072 dom cdm 5618 ‘cfv 6485 (class class class)co 7356 1c1 11030 ℝ^*cxr 11169 ≤ cle 11171 − cmin 11368 ℕ₀^*cxnn0 12501 ..^cfzo 13599 ♯chash 14283 Word cword 14466 iEdgciedg 29084 EdgWalks cewlks 29682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-hash 14284 df-word 14467 df-ewlks 29685

This theorem is referenced by: (None)

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