chnccats1 - Mathbox for Thierry Arnoux

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Theorem chnccats1 32987

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 32979	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
3		chnccats1.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	3	s1cld 14544	. . 3 ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ Word 𝐴)
5		ccatcl 14515	. . 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 2730	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) = (♯‘𝑇))
8	7, 2	wrdfd 14460	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
9	8	fdmd 6680	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 = (0..^(♯‘𝑇)))
10	9	difeq1d 4084	. . . . . . . . 9 ⊢ (𝜑 → (dom 𝑇 ∖ {0}) = ((0..^(♯‘𝑇)) ∖ {0}))
11	10	eleq2d 2814	. . . . . . . 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 32978	. . . . . . . . . 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 3227	. . . . . . 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 14474	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
21	18, 20	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (♯‘𝑇) ∈ ℕ₀)
22	19, 21	elfzodif0 32767	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
23		ccats1val1 14567	. . . . . . 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 3923	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝑇)))
26		ccats1val1 14567	. . . . . . 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 5134	. . . . 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 4297	. . . . . . . 8 ⊢ ¬ 𝑛 ∈ ∅
33		fveq2 6840	. . . . . . . . . . . . . 14 ⊢ (𝑇 = ∅ → (♯‘𝑇) = (♯‘∅))
34		hash0 14308	. . . . . . . . . . . . . 14 ⊢ (♯‘∅) = 0
35	33, 34	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑇 = ∅ → (♯‘𝑇) = 0)
36	35	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → (♯‘𝑇) = 0)
37	36	sneqd 4597	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → {(♯‘𝑇)} = {0})
38	37	difeq1d 4084	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ({0} ∖ {0}))
39		difid 4335	. . . . . . . . . 10 ⊢ ({0} ∖ {0}) = ∅
40	38, 39	eqtrdi 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ∅)
41	40	eleq2d 2814	. . . . . . . 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 3923	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 ∈ {(♯‘𝑇)})
46	45	elsnd 4603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 = (♯‘𝑇))
47	46	oveq1d 7384	. . . . . . . . . 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 6844	. . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ({(♯‘𝑇)} ∖ {0}))
54	53	eldifbd 3924	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ¬ (♯‘𝑇) ∈ {0})
55	52, 54	eldifd 3922	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ (ℕ₀ ∖ {0}))
56		dfn2 12431	. . . . . . . . . . . . 13 ⊢ ℕ = (ℕ₀ ∖ {0})
57	55, 56	eleqtrrdi 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ℕ)
58		fzo0end 13695	. . . . . . . . . . . 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 2828	. . . . . . . . . 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 14505	. . . . . . . . 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 2774	. . . . . . 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 6844	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (♯‘𝑇) = (♯‘𝑇))
70		ccats1val2 14568	. . . . . . . . 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 2764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = 𝑋)
73	44, 65, 72	3brtr4d 5134	. . . . . 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 960	. . . . 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 14561	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘(𝑇 ++ ⟨“𝑋”⟩)) = ((♯‘𝑇) + 1))
79	2, 78	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝑇 ++ ⟨“𝑋”⟩)) = ((♯‘𝑇) + 1))
80	79	eqcomd 2735	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑇) + 1) = (♯‘(𝑇 ++ ⟨“𝑋”⟩)))
81	80, 6	wrdfd 14460	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩):(0..^((♯‘𝑇) + 1))⟶𝐴)
82	81	fdmd 6680	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = (0..^((♯‘𝑇) + 1)))
83	2, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
84		nn0uz 12811	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑇) ∈ (ℤ_≥‘0))
86		fzosplitsn 13712	. . . . . . . . . . 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 2764	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
89	88	difeq1d 4084	. . . . . . . 8 ⊢ (𝜑 → (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) = (((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}) ∖ {0}))
90		difundir 4250	. . . . . . . 8 ⊢ (((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}) ∖ {0}) = (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0}))
91	89, 90	eqtrdi 2780	. . . . . . 7 ⊢ (𝜑 → (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) = (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})))
92	91	eleq2d 2814	. . . . . 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 4112	. . . . 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 960	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
97	96	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
98		ischn 32978	. 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 847 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∖ cdif 3908 ∪ cun 3909 ∅c0 4292 {csn 4585 class class class wbr 5102 dom cdm 5631 ‘cfv 6499 (class class class)co 7369 0cc0 11044 1c1 11045 + caddc 11047 − cmin 11381 ℕcn 12162 ℕ₀cn0 12418 ℤ_≥cuz 12769 ..^cfzo 13591 ♯chash 14271 Word cword 14454 lastSclsw 14503 ++ cconcat 14511 ⟨“cs1 14536 Chaincchn 32976

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-hash 14272 df-word 14455 df-lsw 14504 df-concat 14512 df-s1 14537 df-chn 32977

This theorem is referenced by: constrextdg2lem 33731

W3C validator