wrdpmtrlast - Mathbox for Thierry Arnoux

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Theorem wrdpmtrlast 33174

Description: Reorder a word, so that the symbol given at index 𝐼 is at the end. (Contributed by Thierry Arnoux, 27-May-2025.)

**Hypotheses**
Ref	Expression
wrdpmtrlast.1	⊢ 𝐽 = (0..^(♯‘𝑊))
wrdpmtrlast.2	⊢ (𝜑 → 𝐼 ∈ 𝐽)
wrdpmtrlast.3	⊢ (𝜑 → 𝑊 ∈ Word 𝑆)
wrdpmtrlast.4	⊢ 𝑈 = ((𝑊 ∘ 𝑠) prefix ((♯‘𝑊) − 1))

**Assertion**
Ref	Expression
wrdpmtrlast	⊢ (𝜑 → ∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩)))

Distinct variable groups: 𝐼,𝑠 𝐽,𝑠 𝑊,𝑠 𝜑,𝑠 Allowed substitution hints: 𝑆(𝑠) 𝑈(𝑠)

**Proof of Theorem wrdpmtrlast**
Step	Hyp	Ref	Expression
1		wrdpmtrlast.1	. . 3 ⊢ 𝐽 = (0..^(♯‘𝑊))
2		wrdpmtrlast.2	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐽)
3	1, 2	fzo0pmtrlast 33173	. 2 ⊢ (𝜑 → ∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼))
4		simplr 774	. . . . 5 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑠:𝐽–1-1-onto→𝐽)
5		wrdpmtrlast.4	. . . . . . . 8 ⊢ 𝑈 = ((𝑊 ∘ 𝑠) prefix ((♯‘𝑊) − 1))
6		f1of 6767	. . . . . . . . . . . . . . 15 ⊢ (𝑠:𝐽–1-1-onto→𝐽 → 𝑠:𝐽⟶𝐽)
7	1	feq2i 6647	. . . . . . . . . . . . . . 15 ⊢ (𝑠:𝐽⟶𝐽 ↔ 𝑠:(0..^(♯‘𝑊))⟶𝐽)
8	6, 7	sylib 219	. . . . . . . . . . . . . 14 ⊢ (𝑠:𝐽–1-1-onto→𝐽 → 𝑠:(0..^(♯‘𝑊))⟶𝐽)
9	4, 8	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑠:(0..^(♯‘𝑊))⟶𝐽)
10		iswrdi 14470	. . . . . . . . . . . . 13 ⊢ (𝑠:(0..^(♯‘𝑊))⟶𝐽 → 𝑠 ∈ Word 𝐽)
11	9, 10	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑠 ∈ Word 𝐽)
12		eqidd 2740	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑊) = (♯‘𝑊))
13		wrdpmtrlast.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ Word 𝑆)
14	13	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑊 ∈ Word 𝑆)
15	12, 14	wrdfd 14472	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑊:(0..^(♯‘𝑊))⟶𝑆)
16	1	feq2i 6647	. . . . . . . . . . . . 13 ⊢ (𝑊:𝐽⟶𝑆 ↔ 𝑊:(0..^(♯‘𝑊))⟶𝑆)
17	15, 16	sylibr 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑊:𝐽⟶𝑆)
18		lenco 14785	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Word 𝐽 ∧ 𝑊:𝐽⟶𝑆) → (♯‘(𝑊 ∘ 𝑠)) = (♯‘𝑠))
19	11, 17, 18	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘(𝑊 ∘ 𝑠)) = (♯‘𝑠))
20	9	ffund 6659	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → Fun 𝑠)
21		hashfundm 14395	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ Word 𝐽 ∧ Fun 𝑠) → (♯‘𝑠) = (♯‘dom 𝑠))
22	11, 20, 21	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑠) = (♯‘dom 𝑠))
23	9	fdmd 6665	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → dom 𝑠 = (0..^(♯‘𝑊)))
24	23	fveq2d 6831	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘dom 𝑠) = (♯‘(0..^(♯‘𝑊))))
25	2, 1	eleqtrdi 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
26	25	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝐼 ∈ (0..^(♯‘𝑊)))
27		elfzo0 13646	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
28	27	simp2bi 1152	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) → (♯‘𝑊) ∈ ℕ)
29	26, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑊) ∈ ℕ)
30	29	nnnn0d 12489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑊) ∈ ℕ₀)
31		hashfzo0 14383	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (♯‘(0..^(♯‘𝑊))) = (♯‘𝑊))
32	30, 31	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘(0..^(♯‘𝑊))) = (♯‘𝑊))
33	22, 24, 32	3eqtrd 2778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑠) = (♯‘𝑊))
34	19, 33	eqtr2d 2775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑊) = (♯‘(𝑊 ∘ 𝑠)))
35	34	oveq1d 7371	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) = ((♯‘(𝑊 ∘ 𝑠)) − 1))
36	35	oveq2d 7372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((𝑊 ∘ 𝑠) prefix ((♯‘𝑊) − 1)) = ((𝑊 ∘ 𝑠) prefix ((♯‘(𝑊 ∘ 𝑠)) − 1)))
37	5, 36	eqtrid 2786	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑈 = ((𝑊 ∘ 𝑠) prefix ((♯‘(𝑊 ∘ 𝑠)) − 1)))
38	26	ne0d 4270	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (0..^(♯‘𝑊)) ≠ ∅)
39		f0dom0 6711	. . . . . . . . . . . . 13 ⊢ (𝑠:(0..^(♯‘𝑊))⟶𝐽 → ((0..^(♯‘𝑊)) = ∅ ↔ 𝑠 = ∅))
40	39	necon3bid 2978	. . . . . . . . . . . 12 ⊢ (𝑠:(0..^(♯‘𝑊))⟶𝐽 → ((0..^(♯‘𝑊)) ≠ ∅ ↔ 𝑠 ≠ ∅))
41	40	biimpa 477	. . . . . . . . . . 11 ⊢ ((𝑠:(0..^(♯‘𝑊))⟶𝐽 ∧ (0..^(♯‘𝑊)) ≠ ∅) → 𝑠 ≠ ∅)
42	9, 38, 41	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑠 ≠ ∅)
43		lswco 14792	. . . . . . . . . 10 ⊢ ((𝑠 ∈ Word 𝐽 ∧ 𝑠 ≠ ∅ ∧ 𝑊:𝐽⟶𝑆) → (lastS‘(𝑊 ∘ 𝑠)) = (𝑊‘(lastS‘𝑠)))
44	11, 42, 17, 43	syl3anc 1379	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (lastS‘(𝑊 ∘ 𝑠)) = (𝑊‘(lastS‘𝑠)))
45		lsw 14517	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ Word 𝐽 → (lastS‘𝑠) = (𝑠‘((♯‘𝑠) − 1)))
46	11, 45	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (lastS‘𝑠) = (𝑠‘((♯‘𝑠) − 1)))
47	33	oveq1d 7371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑠) − 1) = ((♯‘𝑊) − 1))
48	47	fveq2d 6831	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑠‘((♯‘𝑠) − 1)) = (𝑠‘((♯‘𝑊) − 1)))
49		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑠‘((♯‘𝑊) − 1)) = 𝐼)
50	46, 48, 49	3eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (lastS‘𝑠) = 𝐼)
51	50	fveq2d 6831	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊‘(lastS‘𝑠)) = (𝑊‘𝐼))
52	44, 51	eqtr2d 2775	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊‘𝐼) = (lastS‘(𝑊 ∘ 𝑠)))
53	52	s1eqd 14555	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ⟨“(𝑊‘𝐼)”⟩ = ⟨“(lastS‘(𝑊 ∘ 𝑠))”⟩)
54	37, 53	oveq12d 7374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩) = (((𝑊 ∘ 𝑠) prefix ((♯‘(𝑊 ∘ 𝑠)) − 1)) ++ ⟨“(lastS‘(𝑊 ∘ 𝑠))”⟩))
55	1, 4, 14	wrdpmcl 33017	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊 ∘ 𝑠) ∈ Word 𝑆)
56		fzo0end 13704	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) ∈ ℕ → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊)))
57	29, 56	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ (0..^(♯‘𝑊)))
58	57, 1	eleqtrrdi 2850	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ 𝐽)
59	17	fdmd 6665	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → dom 𝑊 = 𝐽)
60	58, 59	eleqtrrd 2842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ dom 𝑊)
61		dff1o5 6776	. . . . . . . . . . . . 13 ⊢ (𝑠:𝐽–1-1-onto→𝐽 ↔ (𝑠:𝐽–1-1→𝐽 ∧ ran 𝑠 = 𝐽))
62	61	simprbi 498	. . . . . . . . . . . 12 ⊢ (𝑠:𝐽–1-1-onto→𝐽 → ran 𝑠 = 𝐽)
63	4, 62	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ran 𝑠 = 𝐽)
64	58, 63	eleqtrrd 2842	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ ran 𝑠)
65	60, 64	elind 4129	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ (dom 𝑊 ∩ ran 𝑠))
66	65	ne0d 4270	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (dom 𝑊 ∩ ran 𝑠) ≠ ∅)
67		coeq0 6207	. . . . . . . . 9 ⊢ ((𝑊 ∘ 𝑠) = ∅ ↔ (dom 𝑊 ∩ ran 𝑠) = ∅)
68	67	necon3bii 2986	. . . . . . . 8 ⊢ ((𝑊 ∘ 𝑠) ≠ ∅ ↔ (dom 𝑊 ∩ ran 𝑠) ≠ ∅)
69	66, 68	sylibr 235	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊 ∘ 𝑠) ≠ ∅)
70		pfxlswccat 14666	. . . . . . 7 ⊢ (((𝑊 ∘ 𝑠) ∈ Word 𝑆 ∧ (𝑊 ∘ 𝑠) ≠ ∅) → (((𝑊 ∘ 𝑠) prefix ((♯‘(𝑊 ∘ 𝑠)) − 1)) ++ ⟨“(lastS‘(𝑊 ∘ 𝑠))”⟩) = (𝑊 ∘ 𝑠))
71	55, 69, 70	syl2anc 590	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (((𝑊 ∘ 𝑠) prefix ((♯‘(𝑊 ∘ 𝑠)) − 1)) ++ ⟨“(lastS‘(𝑊 ∘ 𝑠))”⟩) = (𝑊 ∘ 𝑠))
72	54, 71	eqtr2d 2775	. . . . 5 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩))
73	4, 72	jca 516	. . . 4 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩)))
74	73	expl 458	. . 3 ⊢ (𝜑 → ((𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩))))
75	74	eximdv 1924	. 2 ⊢ (𝜑 → (∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩))))
76	3, 75	mpd 15	1 ⊢ (𝜑 → ∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∩ cin 3882 ∅c0 4261 class class class wbr 5072 dom cdm 5618 ran crn 5619 ∘ ccom 5622 Fun wfun 6479 ⟶wf 6481 –1-1→wf1 6482 –1-1-onto→wf1o 6484 ‘cfv 6485 (class class class)co 7356 0cc0 11029 1c1 11030 < clt 11170 − cmin 11368 ℕcn 12165 ℕ₀cn0 12428 ..^cfzo 13599 ♯chash 14283 Word cword 14466 lastSclsw 14515 ++ cconcat 14523 ⟨“cs1 14549 prefix cpfx 14624

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-2o 8396 df-oadd 8399 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9816 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-2 12235 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-lsw 14516 df-concat 14524 df-s1 14550 df-substr 14595 df-pfx 14625 df-pmtr 19408

This theorem is referenced by: 1arithidom 33620

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