lfgrwlkprop - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lfgrwlkprop		Structured version Visualization version GIF version

Theorem lfgrwlkprop 29475

Description: Two adjacent vertices in a walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021.)

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

**Assertion**
Ref	Expression
lfgrwlkprop	⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))

Distinct variable groups: 𝑘,𝐹,𝑥 𝑘,𝐺 𝑘,𝐼,𝑥 𝑃,𝑘 𝑘,𝑉,𝑥 Allowed substitution hints: 𝑃(𝑥) 𝐺(𝑥)

**Proof of Theorem lfgrwlkprop**
Step	Hyp	Ref	Expression
1		wlkv 29400	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
2		eqid 2727	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		lfgrwlkprop.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	2, 3	iswlk 29398	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
5	1, 4	syl 17	. . . 4 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
6		ifptru 1073	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}))
7	6	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹)))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}))
8		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
9		wrdsymbcl 14495	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom 𝐼)
10	9	ad4ant14 751	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹‘𝑘) ∈ dom 𝐼)
11	8, 10	ffvelcdmd 7089	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑘)) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
12		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝐼‘(𝐹‘𝑘)) → (♯‘𝑥) = (♯‘(𝐼‘(𝐹‘𝑘))))
13	12	breq2d 5154	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝐼‘(𝐹‘𝑘)) → (2 ≤ (♯‘𝑥) ↔ 2 ≤ (♯‘(𝐼‘(𝐹‘𝑘)))))
14	13	elrab 3680	. . . . . . . . . . . . . 14 ⊢ ((𝐼‘(𝐹‘𝑘)) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ((𝐼‘(𝐹‘𝑘)) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼‘(𝐹‘𝑘)))))
15		fveq2 6891	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (♯‘(𝐼‘(𝐹‘𝑘))) = (♯‘{(𝑃‘𝑘)}))
16	15	breq2d 5154	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑘))) ↔ 2 ≤ (♯‘{(𝑃‘𝑘)})))
17		fvex 6904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃‘𝑘) ∈ V
18		hashsng 14346	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘𝑘) ∈ V → (♯‘{(𝑃‘𝑘)}) = 1)
19	17, 18	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (♯‘{(𝑃‘𝑘)}) = 1
20	19	breq2i 5150	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ≤ (♯‘{(𝑃‘𝑘)}) ↔ 2 ≤ 1)
21		1lt2 12399	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 2
22		1re 11230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 1 ∈ ℝ
23		2re 12302	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℝ
24	22, 23	ltnlei 11351	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1 < 2 ↔ ¬ 2 ≤ 1)
25		pm2.21 123	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 2 ≤ 1 → (2 ≤ 1 → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
26	24, 25	sylbi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 < 2 → (2 ≤ 1 → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
27	21, 26	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (2 ≤ 1 → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
28	20, 27	sylbi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (2 ≤ (♯‘{(𝑃‘𝑘)}) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
29	16, 28	biimtrdi 252	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (2 ≤ (♯‘(𝐼‘(𝐹‘𝑘))) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
30	29	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (2 ≤ (♯‘(𝐼‘(𝐹‘𝑘))) → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
31	30	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐼‘(𝐹‘𝑘)) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼‘(𝐹‘𝑘)))) → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (((𝐼‘(𝐹‘𝑘)) ∈ 𝒫 𝑉 ∧ 2 ≤ (♯‘(𝐼‘(𝐹‘𝑘)))) → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))))
33	14, 32	biimtrid 241	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((𝐼‘(𝐹‘𝑘)) ∈ {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))))
34	11, 33	mpd 15	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
35	34	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹)))) → ((𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)} → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
36	7, 35	sylbid 239	. . . . . . . . . 10 ⊢ (((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) ∧ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹)))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
37	36	ex 412	. . . . . . . . 9 ⊢ ((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))))
38		neqne 2943	. . . . . . . . . 10 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
39	38	2a1d 26	. . . . . . . . 9 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))))
40	37, 39	pm2.61i 182	. . . . . . . 8 ⊢ ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) ∧ 𝑘 ∈ (0..^(♯‘𝐹))) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
41	40	ralimdva 3162	. . . . . . 7 ⊢ (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
42	41	ex 412	. . . . . 6 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))))
43	42	com23 86	. . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺)) → (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))))
44	43	3impia 1115	. . . 4 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))) → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
45	5, 44	biimtrdi 252	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))))
46	45	pm2.43i 52	. 2 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
47	46	imp 406	1 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 if-wif 1061 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 {crab 3427 Vcvv 3469 ⊆ wss 3944 𝒫 cpw 4598 {csn 4624 {cpr 4626 class class class wbr 5142 dom cdm 5672 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 0cc0 11124 1c1 11125 + caddc 11127 < clt 11264 ≤ cle 11265 2c2 12283 ...cfz 13502 ..^cfzo 13645 ♯chash 14307 Word cword 14482 Vtxcvtx 28783 iEdgciedg 28784 Walkscwlks 29384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-fzo 13646 df-hash 14308 df-word 14483 df-wlks 29387

This theorem is referenced by: lfgriswlk 29476

W3C validator