chnccats1 - Mathbox for Thierry Arnoux

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Theorem chnccats1 33005

Description: Extend a chain with a single element. (Contributed by Thierry Arnoux, 19-Oct-2025.)

**Hypotheses**
Ref	Expression
chnccats1.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
chnccats1.2	⊢ (𝜑 → 𝑇 ∈ ( < Chain𝐴))
chnccats1.3	⊢ (𝜑 → (𝑇 = ∅ ∨ (lastS‘𝑇) < 𝑋))

**Assertion**
Ref	Expression
chnccats1	⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩) ∈ ( < Chain𝐴))

Proof of Theorem chnccats1 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chnccats1.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ( < Chain𝐴))
2	1	chnwrd 32997	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
3		chnccats1.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	3	s1cld 14641	. . 3 ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ Word 𝐴)
5		ccatcl 14612	. . 3 ⊢ ((𝑇 ∈ Word 𝐴 ∧ ⟨“𝑋”⟩ ∈ Word 𝐴) → (𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 584	. 2 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴)
7		eqidd 2738	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) = (♯‘𝑇))
8	7, 2	wrdfd 32918	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
9	8	fdmd 6746	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 = (0..^(♯‘𝑇)))
10	9	difeq1d 4125	. . . . . . . . 9 ⊢ (𝜑 → (dom 𝑇 ∖ {0}) = ((0..^(♯‘𝑇)) ∖ {0}))
11	10	eleq2d 2827	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (dom 𝑇 ∖ {0}) ↔ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})))
12	11	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ (dom 𝑇 ∖ {0}))
13		ischn 32996	. . . . . . . . . 10 ⊢ (𝑇 ∈ ( < Chain𝐴) ↔ (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
14	1, 13	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
15	14	simprd 495	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
16	15	r19.21bi 3251	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom 𝑇 ∖ {0})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
17	12, 16	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
18	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑇 ∈ Word 𝐴)
19		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}))
20		lencl 14571	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
21	18, 20	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (♯‘𝑇) ∈ ℕ₀)
22	19, 21	elfzodif0 32796	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
23		ccats1val1 14664	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
24	18, 22, 23	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
25	19	eldifad 3963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝑇)))
26		ccats1val1 14664	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = (𝑇‘𝑛))
27	18, 25, 26	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = (𝑇‘𝑛))
28	17, 24, 27	3brtr4d 5175	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
29	28	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
30		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
31	30	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
32		noel 4338	. . . . . . . 8 ⊢ ¬ 𝑛 ∈ ∅
33		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑇 = ∅ → (♯‘𝑇) = (♯‘∅))
34		hash0 14406	. . . . . . . . . . . . . 14 ⊢ (♯‘∅) = 0
35	33, 34	eqtrdi 2793	. . . . . . . . . . . . 13 ⊢ (𝑇 = ∅ → (♯‘𝑇) = 0)
36	35	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → (♯‘𝑇) = 0)
37	36	sneqd 4638	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → {(♯‘𝑇)} = {0})
38	37	difeq1d 4125	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ({0} ∖ {0}))
39		difid 4376	. . . . . . . . . 10 ⊢ ({0} ∖ {0}) = ∅
40	38, 39	eqtrdi 2793	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ∅)
41	40	eleq2d 2827	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0}) ↔ 𝑛 ∈ ∅))
42	32, 41	mtbiri 327	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
43	31, 42	pm2.21dd 195	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
44		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (lastS‘𝑇) < 𝑋)
45	30	eldifad 3963	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 ∈ {(♯‘𝑇)})
46	45	elsnd 32547	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 = (♯‘𝑇))
47	46	oveq1d 7446	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (𝑛 − 1) = ((♯‘𝑇) − 1))
48	47	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (𝑛 − 1) = ((♯‘𝑇) − 1))
49	48	fveq2d 6910	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (𝑇‘(𝑛 − 1)) = (𝑇‘((♯‘𝑇) − 1)))
50	2	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → 𝑇 ∈ Word 𝐴)
51	2	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑇 ∈ Word 𝐴)
52	51, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ℕ₀)
53	46, 30	eqeltrrd 2842	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ({(♯‘𝑇)} ∖ {0}))
54	53	eldifbd 3964	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ¬ (♯‘𝑇) ∈ {0})
55	52, 54	eldifd 3962	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ (ℕ₀ ∖ {0}))
56		dfn2 12539	. . . . . . . . . . . . 13 ⊢ ℕ = (ℕ₀ ∖ {0})
57	55, 56	eleqtrrdi 2852	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ℕ)
58		fzo0end 13797	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
59	57, 58	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
60	47, 59	eqeltrd 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
61	60	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
62	50, 61, 23	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
63		lsw 14602	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
64	50, 63	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
65	49, 62, 64	3eqtr4d 2787	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (lastS‘𝑇))
66	46	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → 𝑛 = (♯‘𝑇))
67	66	fveq2d 6910	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)))
68	3	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → 𝑋 ∈ 𝐴)
69		eqidd 2738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (♯‘𝑇) = (♯‘𝑇))
70		ccats1val2 14665	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (♯‘𝑇) = (♯‘𝑇)) → ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)) = 𝑋)
71	50, 68, 69, 70	syl3anc 1373	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)) = 𝑋)
72	67, 71	eqtrd 2777	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = 𝑋)
73	44, 65, 72	3brtr4d 5175	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
74		chnccats1.3	. . . . . . 7 ⊢ (𝜑 → (𝑇 = ∅ ∨ (lastS‘𝑇) < 𝑋))
75	74	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (𝑇 = ∅ ∨ (lastS‘𝑇) < 𝑋))
76	43, 73, 75	mpjaodan 961	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
77	76	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
78		ccatws1len 14658	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘(𝑇 ++ ⟨“𝑋”⟩)) = ((♯‘𝑇) + 1))
79	2, 78	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝑇 ++ ⟨“𝑋”⟩)) = ((♯‘𝑇) + 1))
80	79	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑇) + 1) = (♯‘(𝑇 ++ ⟨“𝑋”⟩)))
81	80, 6	wrdfd 32918	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩):(0..^((♯‘𝑇) + 1))⟶𝐴)
82	81	fdmd 6746	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = (0..^((♯‘𝑇) + 1)))
83	2, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
84		nn0uz 12920	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2851	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑇) ∈ (ℤ_≥‘0))
86		fzosplitsn 13814	. . . . . . . . . . 11 ⊢ ((♯‘𝑇) ∈ (ℤ_≥‘0) → (0..^((♯‘𝑇) + 1)) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
87	85, 86	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0..^((♯‘𝑇) + 1)) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
88	82, 87	eqtrd 2777	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
89	88	difeq1d 4125	. . . . . . . 8 ⊢ (𝜑 → (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) = (((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}) ∖ {0}))
90		difundir 4291	. . . . . . . 8 ⊢ (((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}) ∖ {0}) = (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0}))
91	89, 90	eqtrdi 2793	. . . . . . 7 ⊢ (𝜑 → (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) = (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})))
92	91	eleq2d 2827	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) ↔ 𝑛 ∈ (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0}))))
93	92	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) → 𝑛 ∈ (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})))
94		elun 4153	. . . . 5 ⊢ (𝑛 ∈ (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})) ↔ (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})))
95	93, 94	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})))
96	29, 77, 95	mpjaodan 961	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
97	96	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
98		ischn 32996	. 2 ⊢ ((𝑇 ++ ⟨“𝑋”⟩) ∈ ( < Chain𝐴) ↔ ((𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛)))
99	6, 97, 98	sylanbrc 583	1 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩) ∈ ( < Chain𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∪ cun 3949 ∅c0 4333 {csn 4626 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 − cmin 11492 ℕcn 12266 ℕ₀cn0 12526 ℤ_≥cuz 12878 ..^cfzo 13694 ♯chash 14369 Word cword 14552 lastSclsw 14600 ++ cconcat 14608 ⟨“cs1 14633 Chaincchn 32994

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-chn 32995

This theorem is referenced by: constrextdg2lem 33789

W3C validator