wrdpmtrlast - Mathbox for Thierry Arnoux

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Theorem wrdpmtrlast 33104

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 33103	. 2 ⊢ (𝜑 → ∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼))
4		simplr 768	. . . . 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 6818	. . . . . . . . . . . . . . 15 ⊢ (𝑠:𝐽–1-1-onto→𝐽 → 𝑠:𝐽⟶𝐽)
7	1	feq2i 6698	. . . . . . . . . . . . . . 15 ⊢ (𝑠:𝐽⟶𝐽 ↔ 𝑠:(0..^(♯‘𝑊))⟶𝐽)
8	6, 7	sylib 218	. . . . . . . . . . . . . 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 14535	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑊 ∈ Word 𝑆)
15	12, 14	wrdfd 14537	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑊:(0..^(♯‘𝑊))⟶𝑆)
16	1	feq2i 6698	. . . . . . . . . . . . 13 ⊢ (𝑊:𝐽⟶𝑆 ↔ 𝑊:(0..^(♯‘𝑊))⟶𝑆)
17	15, 16	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑊:𝐽⟶𝑆)
18		lenco 14851	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ Word 𝐽 ∧ 𝑊:𝐽⟶𝑆) → (♯‘(𝑊 ∘ 𝑠)) = (♯‘𝑠))
19	11, 17, 18	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘(𝑊 ∘ 𝑠)) = (♯‘𝑠))
20	9	ffund 6710	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → Fun 𝑠)
21		hashfundm 14460	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ Word 𝐽 ∧ Fun 𝑠) → (♯‘𝑠) = (♯‘dom 𝑠))
22	11, 20, 21	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑠) = (♯‘dom 𝑠))
23	9	fdmd 6716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → dom 𝑠 = (0..^(♯‘𝑊)))
24	23	fveq2d 6880	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘dom 𝑠) = (♯‘(0..^(♯‘𝑊))))
25	2, 1	eleqtrdi 2844	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
26	25	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝐼 ∈ (0..^(♯‘𝑊)))
27		elfzo0 13717	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ (0..^(♯‘𝑊)) ↔ (𝐼 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ ∧ 𝐼 < (♯‘𝑊)))
28	27	simp2bi 1146	. . . . . . . . . . . . . . 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 12562	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑊) ∈ ℕ₀)
31		hashfzo0 14448	. . . . . . . . . . . . 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 2774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑠) = (♯‘𝑊))
34	19, 33	eqtr2d 2771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (♯‘𝑊) = (♯‘(𝑊 ∘ 𝑠)))
35	34	oveq1d 7420	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) = ((♯‘(𝑊 ∘ 𝑠)) − 1))
36	35	oveq2d 7421	. . . . . . . 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 2782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑈 = ((𝑊 ∘ 𝑠) prefix ((♯‘(𝑊 ∘ 𝑠)) − 1)))
38	26	ne0d 4317	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (0..^(♯‘𝑊)) ≠ ∅)
39		f0dom0 6762	. . . . . . . . . . . . 13 ⊢ (𝑠:(0..^(♯‘𝑊))⟶𝐽 → ((0..^(♯‘𝑊)) = ∅ ↔ 𝑠 = ∅))
40	39	necon3bid 2976	. . . . . . . . . . . 12 ⊢ (𝑠:(0..^(♯‘𝑊))⟶𝐽 → ((0..^(♯‘𝑊)) ≠ ∅ ↔ 𝑠 ≠ ∅))
41	40	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑠:(0..^(♯‘𝑊))⟶𝐽 ∧ (0..^(♯‘𝑊)) ≠ ∅) → 𝑠 ≠ ∅)
42	9, 38, 41	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → 𝑠 ≠ ∅)
43		lswco 14858	. . . . . . . . . 10 ⊢ ((𝑠 ∈ Word 𝐽 ∧ 𝑠 ≠ ∅ ∧ 𝑊:𝐽⟶𝑆) → (lastS‘(𝑊 ∘ 𝑠)) = (𝑊‘(lastS‘𝑠)))
44	11, 42, 17, 43	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (lastS‘(𝑊 ∘ 𝑠)) = (𝑊‘(lastS‘𝑠)))
45		lsw 14582	. . . . . . . . . . . 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 7420	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑠) − 1) = ((♯‘𝑊) − 1))
48	47	fveq2d 6880	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑠‘((♯‘𝑠) − 1)) = (𝑠‘((♯‘𝑊) − 1)))
49		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑠‘((♯‘𝑊) − 1)) = 𝐼)
50	46, 48, 49	3eqtrd 2774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (lastS‘𝑠) = 𝐼)
51	50	fveq2d 6880	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊‘(lastS‘𝑠)) = (𝑊‘𝐼))
52	44, 51	eqtr2d 2771	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊‘𝐼) = (lastS‘(𝑊 ∘ 𝑠)))
53	52	s1eqd 14619	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ⟨“(𝑊‘𝐼)”⟩ = ⟨“(lastS‘(𝑊 ∘ 𝑠))”⟩)
54	37, 53	oveq12d 7423	. . . . . 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 32913	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊 ∘ 𝑠) ∈ Word 𝑆)
56		fzo0end 13774	. . . . . . . . . . . . 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 2845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ 𝐽)
59	17	fdmd 6716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → dom 𝑊 = 𝐽)
60	58, 59	eleqtrrd 2837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ dom 𝑊)
61		dff1o5 6827	. . . . . . . . . . . . 13 ⊢ (𝑠:𝐽–1-1-onto→𝐽 ↔ (𝑠:𝐽–1-1→𝐽 ∧ ran 𝑠 = 𝐽))
62	61	simprbi 496	. . . . . . . . . . . 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 2837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ ran 𝑠)
65	60, 64	elind 4175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → ((♯‘𝑊) − 1) ∈ (dom 𝑊 ∩ ran 𝑠))
66	65	ne0d 4317	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (dom 𝑊 ∩ ran 𝑠) ≠ ∅)
67		coeq0 6244	. . . . . . . . 9 ⊢ ((𝑊 ∘ 𝑠) = ∅ ↔ (dom 𝑊 ∩ ran 𝑠) = ∅)
68	67	necon3bii 2984	. . . . . . . 8 ⊢ ((𝑊 ∘ 𝑠) ≠ ∅ ↔ (dom 𝑊 ∩ ran 𝑠) ≠ ∅)
69	66, 68	sylibr 234	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊 ∘ 𝑠) ≠ ∅)
70		pfxlswccat 14731	. . . . . . 7 ⊢ (((𝑊 ∘ 𝑠) ∈ Word 𝑆 ∧ (𝑊 ∘ 𝑠) ≠ ∅) → (((𝑊 ∘ 𝑠) prefix ((♯‘(𝑊 ∘ 𝑠)) − 1)) ++ ⟨“(lastS‘(𝑊 ∘ 𝑠))”⟩) = (𝑊 ∘ 𝑠))
71	55, 69, 70	syl2anc 584	. . . . . 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 2771	. . . . 5 ⊢ (((𝜑 ∧ 𝑠:𝐽–1-1-onto→𝐽) ∧ (𝑠‘((♯‘𝑊) − 1)) = 𝐼) → (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩))
73	4, 72	jca 511	. . . 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 457	. . 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 1917	. 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 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2932 ∩ cin 3925 ∅c0 4308 class class class wbr 5119 dom cdm 5654 ran crn 5655 ∘ ccom 5658 Fun wfun 6525 ⟶wf 6527 –1-1→wf1 6528 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 0cc0 11129 1c1 11130 < clt 11269 − cmin 11466 ℕcn 12240 ℕ₀cn0 12501 ..^cfzo 13671 ♯chash 14348 Word cword 14531 lastSclsw 14580 ++ cconcat 14588 ⟨“cs1 14613 prefix cpfx 14688

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-pmtr 19423

This theorem is referenced by: 1arithidom 33552

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