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Theorem chnccats1 18552

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 18535	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
3		chnccats1.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	3	s1cld 14531	. . 3 ⊢ (𝜑 → ⟨“𝑋”⟩ ∈ Word 𝐴)
5		ccatcl 14501	. . 3 ⊢ ((𝑇 ∈ Word 𝐴 ∧ ⟨“𝑋”⟩ ∈ Word 𝐴) → (𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 585	. 2 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩) ∈ Word 𝐴)
7		eqidd 2738	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) = (♯‘𝑇))
8	7, 2	wrdfd 14446	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
9	8	fdmd 6673	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 = (0..^(♯‘𝑇)))
10	9	difeq1d 4078	. . . . . . . . 9 ⊢ (𝜑 → (dom 𝑇 ∖ {0}) = ((0..^(♯‘𝑇)) ∖ {0}))
11	10	eleq2d 2823	. . . . . . . 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 18534	. . . . . . . . . 10 ⊢ (𝑇 ∈ ( < Chain 𝐴) ↔ (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 3229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom 𝑇 ∖ {0})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
17	12, 16	syldan 592	. . . . . 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 14460	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
21	18, 20	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (♯‘𝑇) ∈ ℕ₀)
22	19, 21	elfzodif0 13690	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
23		ccats1val1 14554	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
24	18, 22, 23	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
25	19	eldifad 3914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝑇)))
26		ccats1val1 14554	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = (𝑇‘𝑛))
27	18, 25, 26	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = (𝑇‘𝑛))
28	17, 24, 27	3brtr4d 5131	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
29	28	adantlr 716	. . . 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 4291	. . . . . . . 8 ⊢ ¬ 𝑛 ∈ ∅
33		fveq2 6835	. . . . . . . . . . . . . 14 ⊢ (𝑇 = ∅ → (♯‘𝑇) = (♯‘∅))
34		hash0 14294	. . . . . . . . . . . . . 14 ⊢ (♯‘∅) = 0
35	33, 34	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ (𝑇 = ∅ → (♯‘𝑇) = 0)
36	35	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → (♯‘𝑇) = 0)
37	36	sneqd 4593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → {(♯‘𝑇)} = {0})
38	37	difeq1d 4078	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ({0} ∖ {0}))
39		difid 4329	. . . . . . . . . 10 ⊢ ({0} ∖ {0}) = ∅
40	38, 39	eqtrdi 2788	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0}) = ∅)
41	40	eleq2d 2823	. . . . . . . 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 3914	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 ∈ {(♯‘𝑇)})
46	45	elsnd 4599	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → 𝑛 = (♯‘𝑇))
47	46	oveq1d 7375	. . . . . . . . . 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 6839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → (𝑇‘(𝑛 − 1)) = (𝑇‘((♯‘𝑇) − 1)))
50	2	ad2antrr 727	. . . . . . . . 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 2838	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ({(♯‘𝑇)} ∖ {0}))
54	53	eldifbd 3915	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ¬ (♯‘𝑇) ∈ {0})
55	52, 54	eldifd 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ (ℕ₀ ∖ {0}))
56		dfn2 12418	. . . . . . . . . . . . 13 ⊢ ℕ = (ℕ₀ ∖ {0})
57	55, 56	eleqtrrdi 2848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → (♯‘𝑇) ∈ ℕ)
58		fzo0end 13678	. . . . . . . . . . . 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 2837	. . . . . . . . . 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 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
63		lsw 14491	. . . . . . . . 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 2782	. . . . . . 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 6839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)))
68	3	ad2antrr 727	. . . . . . . . 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 14555	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (♯‘𝑇) = (♯‘𝑇)) → ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)) = 𝑋)
71	50, 68, 69, 70	syl3anc 1374	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘(♯‘𝑇)) = 𝑋)
72	67, 71	eqtrd 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) ∧ (lastS‘𝑇) < 𝑋) → ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛) = 𝑋)
73	44, 65, 72	3brtr4d 5131	. . . . . 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 716	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})) ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0})) → ((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
78		ccatws1len 14548	. . . . . . . . . . . . . 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 14446	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩):(0..^((♯‘𝑇) + 1))⟶𝐴)
82	81	fdmd 6673	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = (0..^((♯‘𝑇) + 1)))
83	2, 20	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
84		nn0uz 12793	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
85	83, 84	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑇) ∈ (ℤ_≥‘0))
86		fzosplitsn 13696	. . . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑇 ++ ⟨“𝑋”⟩) = ((0..^(♯‘𝑇)) ∪ {(♯‘𝑇)}))
89	88	difeq1d 4078	. . . . . . . 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 2788	. . . . . . 7 ⊢ (𝜑 → (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0}) = (((0..^(♯‘𝑇)) ∖ {0}) ∪ ({(♯‘𝑇)} ∖ {0})))
92	91	eleq2d 2823	. . . . . 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 4106	. . . . 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 3129	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom (𝑇 ++ ⟨“𝑋”⟩) ∖ {0})((𝑇 ++ ⟨“𝑋”⟩)‘(𝑛 − 1)) < ((𝑇 ++ ⟨“𝑋”⟩)‘𝑛))
98		ischn 18534	. 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 584	1 ⊢ (𝜑 → (𝑇 ++ ⟨“𝑋”⟩) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∖ cdif 3899 ∪ cun 3900 ∅c0 4286 {csn 4581 class class class wbr 5099 dom cdm 5625 ‘cfv 6493 (class class class)co 7360 0cc0 11030 1c1 11031 + caddc 11033 − cmin 11368 ℕcn 12149 ℕ₀cn0 12405 ℤ_≥cuz 12755 ..^cfzo 13574 ♯chash 14257 Word cword 14440 lastSclsw 14489 ++ cconcat 14497 ⟨“cs1 14523 Chain cchn 18532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-lsw 14490 df-concat 14498 df-s1 14524 df-chn 18533

This theorem is referenced by: constrextdg2lem 33907 nthrucw 47197

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