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Theorem ewlkinedg 29693

Description: The intersection (common vertices) of two adjacent edges in an s-walk of edges. (Contributed by AV, 4-Jan-2021.)

**Hypothesis**
Ref	Expression
ewlksfval.i	⊢ 𝐼 = (iEdg‘𝐺)

**Assertion**
Ref	Expression
ewlkinedg	⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝐾 ∈ (1..^(♯‘𝐹))) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾)))))

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

Step	Hyp	Ref	Expression
1		ewlksfval.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	ewlkprop 29692	. . 3 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))
3		fvoveq1 7381	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝐹‘(𝑘 − 1)) = (𝐹‘(𝐾 − 1)))
4	3	fveq2d 6836	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝐼‘(𝐹‘(𝑘 − 1))) = (𝐼‘(𝐹‘(𝐾 − 1))))
5		2fveq3 6837	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝐼‘(𝐹‘𝑘)) = (𝐼‘(𝐹‘𝐾)))
6	4, 5	ineq12d 4162	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))) = ((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾))))
7	6	fveq2d 6836	. . . . . 6 ⊢ (𝑘 = 𝐾 → (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))) = (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾)))))
8	7	breq2d 5098	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))) ↔ 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾))))))
9	8	rspccv 3562	. . . 4 ⊢ (∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))) → (𝐾 ∈ (1..^(♯‘𝐹)) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾))))))
10	9	3ad2ant3 1136	. . 3 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ ℕ₀^*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))) → (𝐾 ∈ (1..^(♯‘𝐹)) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾))))))
11	2, 10	syl 17	. 2 ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → (𝐾 ∈ (1..^(♯‘𝐹)) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾))))))
12	11	imp 406	1 ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝐾 ∈ (1..^(♯‘𝐹))) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ∩ cin 3889 class class class wbr 5086 dom cdm 5622 ‘cfv 6490 (class class class)co 7358 1c1 11028 ≤ cle 11169 − cmin 11366 ℕ₀^*cxnn0 12499 ..^cfzo 13597 ♯chash 14281 Word cword 14464 iEdgciedg 29085 EdgWalks cewlks 29684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-n0 12427 df-z 12514 df-uz 12778 df-fz 13451 df-fzo 13598 df-hash 14282 df-word 14465 df-ewlks 29687

This theorem is referenced by: (None)

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