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Theorem wlkp1lem1 29757

Description: Lemma for wlkp1 29765. (Contributed by AV, 6-Mar-2021.)

**Hypotheses**
Ref	Expression
wlkp1.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkp1.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkp1.f	⊢ (𝜑 → Fun 𝐼)
wlkp1.a	⊢ (𝜑 → 𝐼 ∈ Fin)
wlkp1.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
wlkp1.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
wlkp1.d	⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
wlkp1.w	⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
wlkp1.n	⊢ 𝑁 = (♯‘𝐹)

**Assertion**
Ref	Expression
wlkp1lem1	⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)

**Proof of Theorem wlkp1lem1**
Step	Hyp	Ref	Expression
1		wlkp1.w	. 2 ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
2		wlkcl 29701	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
3		wlkp1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	wlkp 29702	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
5	2, 4	jca 511	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → ((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉))
6		fzp1nel 13539	. . . . . 6 ⊢ ¬ ((♯‘𝐹) + 1) ∈ (0...(♯‘𝐹))
7	6	a1i 11	. . . . 5 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ¬ ((♯‘𝐹) + 1) ∈ (0...(♯‘𝐹)))
8		wlkp1.n	. . . . . . 7 ⊢ 𝑁 = (♯‘𝐹)
9	8	oveq1i 7378	. . . . . 6 ⊢ (𝑁 + 1) = ((♯‘𝐹) + 1)
10	9	eleq1i 2828	. . . . 5 ⊢ ((𝑁 + 1) ∈ (0...(♯‘𝐹)) ↔ ((♯‘𝐹) + 1) ∈ (0...(♯‘𝐹)))
11	7, 10	sylnibr 329	. . . 4 ⊢ ((♯‘𝐹) ∈ ℕ₀ → ¬ (𝑁 + 1) ∈ (0...(♯‘𝐹)))
12		eleq2 2826	. . . . 5 ⊢ (dom 𝑃 = (0...(♯‘𝐹)) → ((𝑁 + 1) ∈ dom 𝑃 ↔ (𝑁 + 1) ∈ (0...(♯‘𝐹))))
13	12	notbid 318	. . . 4 ⊢ (dom 𝑃 = (0...(♯‘𝐹)) → (¬ (𝑁 + 1) ∈ dom 𝑃 ↔ ¬ (𝑁 + 1) ∈ (0...(♯‘𝐹))))
14	11, 13	syl5ibrcom 247	. . 3 ⊢ ((♯‘𝐹) ∈ ℕ₀ → (dom 𝑃 = (0...(♯‘𝐹)) → ¬ (𝑁 + 1) ∈ dom 𝑃))
15		fdm 6679	. . 3 ⊢ (𝑃:(0...(♯‘𝐹))⟶𝑉 → dom 𝑃 = (0...(♯‘𝐹)))
16	14, 15	impel 505	. 2 ⊢ (((♯‘𝐹) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) → ¬ (𝑁 + 1) ∈ dom 𝑃)
17	1, 5, 16	3syl 18	1 ⊢ (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 dom cdm 5632 Fun wfun 6494 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 0cc0 11038 1c1 11039 + caddc 11041 ℕ₀cn0 12413 ...cfz 13435 ♯chash 14265 Vtxcvtx 29081 iEdgciedg 29082 Walkscwlks 29682

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 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 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-wlks 29685

This theorem is referenced by: wlkp1lem7 29763 wlkp1lem8 29764 eupth2eucrct 30304

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