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Theorem chnccats1 18685

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 ∈ ( < Chain 𝐴))
chnccats1.3	⊢ (𝜑 → (𝑇 = ∅ ∨ (lastS‘𝑇) < 𝑋))

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

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

Step	Hyp	Ref	Expression
1		chnccats1.2	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ( < Chain 𝐴))
2	1	chnwrd 18668	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
3		chnccats1.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	3	s1cld 14646	. . 3 ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ Word 𝐴)
5		ccatcl 14616	. . 3 ⊢ ((𝑇 ∈ Word 𝐴 ∧ ⟨“𝑋”⟩ ∈ Word 𝐴) → (𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴)
7		eqidd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) = (♯‘𝑇))
8	7, 2	wrdfd 14561	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
9	8	fdmd 6716	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 = (0..^(♯‘𝑇)))
10	9	difeq1d 4080	. . . . . . . . 9 ⊢ (𝜑 → (dom 𝑇 ∖ {0}) = ((0..^(♯‘𝑇)) ∖ {0}))
11	10	eleq2d 2849	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (dom 𝑇 ∖ {0}) ↔ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})))
12	11	biimpar 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ (dom 𝑇 ∖ {0}))
13		ischn 18667	. . . . . . . . . 10 ⊢ (𝑇 ∈ ( < Chain 𝐴) ↔ (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
14	1, 13	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
15	14	simprd 500	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
16	15	r19.21bi 3257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom 𝑇 ∖ {0})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
17	12, 16	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
18	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑇 ∈ Word 𝐴)
19		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}))
20		lencl 14575	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
21	18, 20	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (♯‘𝑇) ∈ ℕ₀)
22	19, 21	elfzodif0 13804	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
23		ccats1val1 14669	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
24	18, 22, 23	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
25	19	eldifad 3917	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝑇)))
26		ccats1val1 14669	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = (𝑇‘𝑛))
27	18, 25, 26	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = (𝑇‘𝑛))
28	17, 24, 27	3brtr4d 5143	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
29	28	adantlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
30		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
31	30	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
32		noel 4291	. . . . . . . 8 ⊢ ¬ 𝑛 ∈ ∅
33		fveq2 6881	. . . . . . . . . . . . . 14 ⊢ (𝑇 = ∅ → (♯‘𝑇) = (♯‘∅))
34		hash0 14408	. . . . . . . . . . . . . 14 ⊢ (♯‘∅) = 0
35	33, 34	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑇 = ∅ → (♯‘𝑇) = 0)
36	35	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → (♯‘𝑇) = 0)
37	36	sneqd 4601	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → {(♯‘𝑇)} = {0})
38	37	difeq1d 4080	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ({0} ∖ {0}))
39		difid 4332	. . . . . . . . . 10 ⊢ ({0} ∖ {0}) = ∅
40	38, 39	eqtrdi 2814	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ∅)
41	40	eleq2d 2849	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0}) ↔ 𝑛 ∈ ∅))
42	32, 41	mtbiri 330	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
43	31, 42	pm2.21dd 198	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
44		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (lastS‘𝑇) < 𝑋)
45	30	eldifad 3917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 ∈ {(♯‘𝑇)})
46	45	elsnd 4607	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 = (♯‘𝑇))
47	46	oveq1d 7425	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (𝑛 − 1) = ((♯‘𝑇) − 1))
48	47	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (𝑛 − 1) = ((♯‘𝑇) − 1))
49	48	fveq2d 6885	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (𝑇‘(𝑛 − 1)) = (𝑇‘((♯‘𝑇) − 1)))
50	2	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → 𝑇 ∈ Word 𝐴)
51	2	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑇 ∈ Word 𝐴)
52	51, 20	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ℕ₀)
53	46, 30	eqeltrrd 2864	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ({(♯‘𝑇)} ∖ {0}))
54	53	eldifbd 3918	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ¬ (♯‘𝑇) ∈ {0})
55	52, 54	eldifd 3916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ (ℕ₀ ∖ {0}))
56		dfn2 12521	. . . . . . . . . . . . 13 ⊢ ℕ = (ℕ₀ ∖ {0})
57	55, 56	eleqtrrdi 2874	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ℕ)
58		fzo0end 13792	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
59	57, 58	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
60	47, 59	eqeltrd 2863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
61	60	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
62	50, 61, 23	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
63		lsw 14606	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
64	50, 63	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
65	49, 62, 64	3eqtr4d 2808	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (lastS‘𝑇))
66	46	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → 𝑛 = (♯‘𝑇))
67	66	fveq2d 6885	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)))
68	3	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → 𝑋 ∈ 𝐴)
69		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (♯‘𝑇) = (♯‘𝑇))
70		ccats1val2 14670	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (♯‘𝑇) = (♯‘𝑇)) → ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)) = 𝑋)
71	50, 68, 69, 70	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)) = 𝑋)
72	67, 71	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = 𝑋)
73	44, 65, 72	3brtr4d 5143	. . . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (𝑇 = ∅ ∨ (lastS‘𝑇) < 𝑋))
76	43, 73, 75	mpjaodan 973	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
77	76	adantlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
78		ccatws1len 14663	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘(𝑇 ++ ⟨“𝑋”⟩)) = ((♯‘𝑇) + 1))
79	2, 78	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝑇 ++ ⟨“𝑋”⟩)) = ((♯‘𝑇) + 1))
80	79	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑇) + 1) = (♯‘(𝑇 ++ ⟨“𝑋”⟩)))
81	80, 6	wrdfd 14561	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩):(0..^((♯‘𝑇) + 1))⟶𝐴)
82	81	fdmd 6716	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = (0..^((♯‘𝑇) + 1)))
83	2, 20	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
84		nn0uz 12904	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2873	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑇) ∈ (ℤ_≥‘0))
86		fzosplitsn 13810	. . . . . . . . . . 11 ⊢ ((♯‘𝑇) ∈ (ℤ_≥‘0) → (0..^((♯‘𝑇) + 1)) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
87	85, 86	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0..^((♯‘𝑇) + 1)) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
88	82, 87	eqtrd 2798	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
89	88	difeq1d 4080	. . . . . . . 8 ⊢ (𝜑 → (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) = (((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}) ∖ {0}))
90		difundir 4244	. . . . . . . 8 ⊢ (((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}) ∖ {0}) = (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0}))
91	89, 90	eqtrdi 2814	. . . . . . 7 ⊢ (𝜑 → (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) = (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})))
92	91	eleq2d 2849	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) ↔ 𝑛 ∈ (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0}))))
93	92	biimpa 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) → 𝑛 ∈ (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})))
94		elun 4107	. . . . 5 ⊢ (𝑛 ∈ (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})) ↔ (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})))
95	93, 94	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})))
96	29, 77, 95	mpjaodan 973	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
97	96	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
98		ischn 18667	. 2 ⊢ ((𝑇 ++ ⟨“𝑋”⟩) ∈ ( < Chain 𝐴) ↔ ((𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛)))
99	6, 97, 98	sylanbrc 594	1 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∖ cdif 3902 ∪ cun 3903 ∅c0 4286 {csn 4589 class class class wbr 5109 dom cdm 5661 ‘cfv 6536 (class class class)co 7410 0cc0 11104 1c1 11105 + caddc 11107 − cmin 11445 ℕcn 12237 ℕ₀cn0 12508 ℤ_≥cuz 12866 ..^cfzo 13687 ♯chash 14371 Word cword 14555 lastSclsw 14604 ++ cconcat 14612 ⟨“cs1 14638 Chain cchn 18665

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-n0 12509 df-z 12596 df-uz 12867 df-fz 13540 df-fzo 13688 df-hash 14372 df-word 14556 df-lsw 14605 df-concat 14613 df-s1 14639 df-chn 18666

This theorem is used by: constrextdg2lem 34147 nthrucw 47635

W3C validator