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Theorem chnccat 18668

Description: Concatenate two chains. (Contributed by Ender Ting, 20-Jan-2026.)

**Hypotheses**
Ref	Expression
chnccat.1	⊢ (𝜑 → 𝑇 ∈ ( < Chain 𝐴))
chnccat.2	⊢ (𝜑 → 𝑈 ∈ ( < Chain 𝐴))
chnccat.3	⊢ (𝜑 → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0)))

**Assertion**
Ref	Expression
chnccat	⊢ (𝜑 → (𝑇 ++ 𝑈) ∈ ( < Chain 𝐴))

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

Step	Hyp	Ref	Expression
1		chnccat.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ( < Chain 𝐴))
2	1	chnwrd 18650	. . 3 ⊢ (𝜑 → 𝑇 ∈ Word 𝐴)
3		chnccat.2	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ( < Chain 𝐴))
4	3	chnwrd 18650	. . 3 ⊢ (𝜑 → 𝑈 ∈ Word 𝐴)
5		ccatcl 14597	. . 3 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴) → (𝑇 ++ 𝑈) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 593	. 2 ⊢ (𝜑 → (𝑇 ++ 𝑈) ∈ Word 𝐴)
7		eqidd 2764	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) = (♯‘𝑇))
8	7, 2	wrdfd 14542	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
9	8	fdmd 6702	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑇 = (0..^(♯‘𝑇)))
10	9	difeq1d 4080	. . . . . . . . 9 ⊢ (𝜑 → (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
11	10	eleq2d 2849	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
12	11	biimpar 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
13		snsspr1 4773	. . . . . . . . . 10 ⊢ {0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))}
14		sscon 4097	. . . . . . . . . 10 ⊢ ({0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} → (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ (dom 𝑇 ∖ {0}))
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ (dom 𝑇 ∖ {0})
16	15	sseli 3933	. . . . . . . 8 ⊢ (𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ (dom 𝑇 ∖ {0}))
17		ischn 18649	. . . . . . . . . . 11 ⊢ (𝑇 ∈ ( < Chain 𝐴) ↔ (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
18	1, 17	sylib 220	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛)))
19	18	simprd 499	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ (dom 𝑇 ∖ {0})(𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
20	19	r19.21bi 3255	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom 𝑇 ∖ {0})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
21	16, 20	sylan2 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom 𝑇 ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
22	12, 21	syldan 600	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) < (𝑇‘𝑛))
23	2	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑇 ∈ Word 𝐴)
24	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ Word 𝐴)
25		sscon 4097	. . . . . . . . . . 11 ⊢ ({0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} → ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ((0..^(♯‘𝑇)) ∖ {0}))
26	13, 25	ax-mp 5	. . . . . . . . . 10 ⊢ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ((0..^(♯‘𝑇)) ∖ {0})
27	26	sseli 3933	. . . . . . . . 9 ⊢ (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}))
28	27	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0}))
29		lencl 14556	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
30	2, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
31	30	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℕ₀)
32	28, 31	elfzodif0 13786	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
33		ccatval1 14600	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
34	23, 24, 32, 33	syl3anc 1392	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
35		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
36	35	eldifad 3917	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ (0..^(♯‘𝑇)))
37		ccatval1 14600	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑇‘𝑛))
38	23, 24, 36, 37	syl3anc 1392	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑇‘𝑛))
39	22, 34, 38	3brtr4d 5133	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
40	39	adantlr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) ∧ 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
41		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
42	41	adantr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
43		noel 4291	. . . . . . . 8 ⊢ ¬ 𝑛 ∈ ∅
44		fveq2 6867	. . . . . . . . . . . . . 14 ⊢ (𝑇 = ∅ → (♯‘𝑇) = (♯‘∅))
45		hash0 14390	. . . . . . . . . . . . . 14 ⊢ (♯‘∅) = 0
46	44, 45	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑇 = ∅ → (♯‘𝑇) = 0)
47	46	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → (♯‘𝑇) = 0)
48	47	sneqd 4595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → {(♯‘𝑇)} = {0})
49	48	difeq1d 4080	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ({0} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
50		difpr 4764	. . . . . . . . . . 11 ⊢ ({0} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = (({0} ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))})
51		difid 4330	. . . . . . . . . . . 12 ⊢ ({0} ∖ {0}) = ∅
52	51	difeq1i 4077	. . . . . . . . . . 11 ⊢ (({0} ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}) = (∅ ∖ {((♯‘𝑇) + (♯‘𝑈))})
53		0dif 4360	. . . . . . . . . . 11 ⊢ (∅ ∖ {((♯‘𝑇) + (♯‘𝑈))}) = ∅
54	50, 52, 53	3eqtri 2790	. . . . . . . . . 10 ⊢ ({0} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅
55	49, 54	eqtrdi 2814	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅)
56	55	eleq2d 2849	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ 𝑛 ∈ ∅))
57	43, 56	mtbiri 329	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
58	42, 57	pm2.21dd 197	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑇 = ∅) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
59		eldifi 4085	. . . . . . . . 9 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ {(♯‘𝑇)})
60	59	elsnd 4601	. . . . . . . 8 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 = (♯‘𝑇))
61	60	ad2antlr 737	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → 𝑛 = (♯‘𝑇))
62		vex 3459	. . . . . . . . 9 ⊢ 𝑛 ∈ V
63	62	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → 𝑛 ∈ V)
64		eldifn 4086	. . . . . . . . . 10 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
65	64	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
66		fveq2 6867	. . . . . . . . . . . . . 14 ⊢ (𝑈 = ∅ → (♯‘𝑈) = (♯‘∅))
67	66, 45	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝑈 = ∅ → (♯‘𝑈) = 0)
68	67	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → (♯‘𝑈) = 0)
69	68	oveq2d 7412	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((♯‘𝑇) + (♯‘𝑈)) = ((♯‘𝑇) + 0))
70		nn0cn 12501	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℕ₀ → (♯‘𝑇) ∈ ℂ)
71	2, 29, 70	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑇) ∈ ℂ)
72	71	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℂ)
73	72	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → (♯‘𝑇) ∈ ℂ)
74	73	addridd 11394	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((♯‘𝑇) + 0) = (♯‘𝑇))
75	69, 74	eqtrd 2798	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((♯‘𝑇) + (♯‘𝑈)) = (♯‘𝑇))
76	75	preq2d 4700	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → {0, ((♯‘𝑇) + (♯‘𝑈))} = {0, (♯‘𝑇)})
77	65, 76	neleqtrd 2885	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ¬ 𝑛 ∈ {0, (♯‘𝑇)})
78	63, 77	nelpr2 4613	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → 𝑛 ≠ (♯‘𝑇))
79	61, 78	pm2.21ddne 3042	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑈 = ∅) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
80		simpr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (lastS‘𝑇) < (𝑈‘0))
81	2	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑇 ∈ Word 𝐴)
82	81	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 𝑇 ∈ Word 𝐴)
83	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ Word 𝐴)
84	83	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 𝑈 ∈ Word 𝐴)
85	41	eldifad 3917	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ {(♯‘𝑇)})
86	85	elsnd 4601	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 = (♯‘𝑇))
87	86	oveq1d 7411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) = ((♯‘𝑇) − 1))
88	81, 29	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℕ₀)
89		sscon 4097	. . . . . . . . . . . . . . . . . . 19 ⊢ ({0} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ({(♯‘𝑇)} ∖ {0}))
90	13, 89	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ⊆ ({(♯‘𝑇)} ∖ {0})
91	90	sseli 3933	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0}))
92	60, 91	eqeltrrd 2864	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → (♯‘𝑇) ∈ ({(♯‘𝑇)} ∖ {0}))
93	92	eldifbd 3918	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → ¬ (♯‘𝑇) ∈ {0})
94	93	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ¬ (♯‘𝑇) ∈ {0})
95	88, 94	eldifd 3916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ (ℕ₀ ∖ {0}))
96		dfn2 12504	. . . . . . . . . . . . 13 ⊢ ℕ = (ℕ₀ ∖ {0})
97	95, 96	eleqtrrdi 2874	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℕ)
98		fzo0end 13774	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
99	97, 98	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((♯‘𝑇) − 1) ∈ (0..^(♯‘𝑇)))
100	87, 99	eqeltrd 2863	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
101	100	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (𝑛 − 1) ∈ (0..^(♯‘𝑇)))
102	82, 84, 101, 33	syl3anc 1392	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑇‘(𝑛 − 1)))
103	60	oveq1d 7411	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → (𝑛 − 1) = ((♯‘𝑇) − 1))
104	103	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) = ((♯‘𝑇) − 1))
105	104	fveq2d 6871	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) = (𝑇‘((♯‘𝑇) − 1)))
106		lsw 14587	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
107	81, 106	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (lastS‘𝑇) = (𝑇‘((♯‘𝑇) − 1)))
108	105, 107	eqtr4d 2801	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇‘(𝑛 − 1)) = (lastS‘𝑇))
109	108	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (𝑇‘(𝑛 − 1)) = (lastS‘𝑇))
110	102, 109	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (lastS‘𝑇))
111	86	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 𝑛 = (♯‘𝑇))
112	111	fveq2d 6871	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘𝑛) = ((𝑇 ++ 𝑈)‘(♯‘𝑇)))
113		lencl 14556	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ Word 𝐴 → (♯‘𝑈) ∈ ℕ₀)
114	4, 113	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑈) ∈ ℕ₀)
115	114	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ∈ ℕ₀)
116	81	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → 𝑇 ∈ Word 𝐴)
117	116, 29, 70	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → (♯‘𝑇) ∈ ℂ)
118		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → (♯‘𝑈) = 0)
119	117, 118	jca 519	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0))
120		prid2g 4721	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑇) ∈ ℂ → (♯‘𝑇) ∈ {0, (♯‘𝑇)})
121	120	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → (♯‘𝑇) ∈ {0, (♯‘𝑇)})
122		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → (♯‘𝑈) = 0)
123	122	oveq2d 7412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) + (♯‘𝑈)) = ((♯‘𝑇) + 0))
124		addrid 11374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑇) ∈ ℂ → ((♯‘𝑇) + 0) = (♯‘𝑇))
125	124	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) + 0) = (♯‘𝑇))
126	123, 125	eqtrd 2798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ((♯‘𝑇) + (♯‘𝑈)) = (♯‘𝑇))
127	126	preq2d 4700	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → {0, ((♯‘𝑇) + (♯‘𝑈))} = {0, (♯‘𝑇)})
128	121, 127	eleqtrrd 2866	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → (♯‘𝑇) ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
129	128	snssd 4746	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → {(♯‘𝑇)} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))})
130		ssdif0 4320	. . . . . . . . . . . . . . . . 17 ⊢ ({(♯‘𝑇)} ⊆ {0, ((♯‘𝑇) + (♯‘𝑈))} ↔ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅)
131	129, 130	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑈) = 0) → ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅)
132		nel02 4292	. . . . . . . . . . . . . . . 16 ⊢ (({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = ∅ → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
133	119, 131, 132	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (♯‘𝑈) = 0) → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
134	133	ex 416	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((♯‘𝑈) = 0 → ¬ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
135	41, 134	mt2d 136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ¬ (♯‘𝑈) = 0)
136	135	neqned 2965	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ≠ 0)
137		elnnne0 12505	. . . . . . . . . . . 12 ⊢ ((♯‘𝑈) ∈ ℕ ↔ ((♯‘𝑈) ∈ ℕ₀ ∧ (♯‘𝑈) ≠ 0))
138	115, 136, 137	sylanbrc 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ∈ ℕ)
139	138	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → (♯‘𝑈) ∈ ℕ)
140		lbfzo0 13715	. . . . . . . . . 10 ⊢ (0 ∈ (0..^(♯‘𝑈)) ↔ (♯‘𝑈) ∈ ℕ)
141	139, 140	sylibr 236	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → 0 ∈ (0..^(♯‘𝑈)))
142		addlid 11377	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℂ → (0 + (♯‘𝑇)) = (♯‘𝑇))
143	142	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑇) ∈ ℂ → (♯‘𝑇) = (0 + (♯‘𝑇)))
144	143	fveq2d 6871	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℂ → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))))
145	29, 70, 144	3syl 18	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))))
146	145	3ad2ant1 1147	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 0 ∈ (0..^(♯‘𝑈))) → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))))
147		ccatval3 14602	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 0 ∈ (0..^(♯‘𝑈))) → ((𝑇 ++ 𝑈)‘(0 + (♯‘𝑇))) = (𝑈‘0))
148	146, 147	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 0 ∈ (0..^(♯‘𝑈))) → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = (𝑈‘0))
149	82, 84, 141, 148	syl3anc 1392	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(♯‘𝑇)) = (𝑈‘0))
150	112, 149	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑈‘0))
151	80, 110, 150	3brtr4d 5133	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ (lastS‘𝑇) < (𝑈‘0)) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
152		chnccat.3	. . . . . . 7 ⊢ (𝜑 → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0)))
153	152	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0)))
154	58, 79, 151, 153	mpjao3dan 1453	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
155	154	adantlr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) ∧ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
156		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
157		eldifi 4085	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}))
158	157	eldifad 3917	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
159		elfzoelz 13674	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) → 𝑛 ∈ ℤ)
160	158, 159	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) → 𝑛 ∈ ℤ)
161		zcn 12583	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
162	156, 160, 161	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ℂ)
163		1cnd 11186	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 1 ∈ ℂ)
164	71	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑇) ∈ ℂ)
165	162, 163, 164	sub32d 11585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑛 − 1) − (♯‘𝑇)) = ((𝑛 − (♯‘𝑇)) − 1))
166	165	fveq2d 6871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑈‘((𝑛 − 1) − (♯‘𝑇))) = (𝑈‘((𝑛 − (♯‘𝑇)) − 1)))
167	3	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ ( < Chain 𝐴))
168	158	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
169		nn0z 12602	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑈) ∈ ℕ₀ → (♯‘𝑈) ∈ ℤ)
170	4, 113, 169	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑈) ∈ ℤ)
171	170	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (♯‘𝑈) ∈ ℤ)
172		fzosubel3 13742	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑈) ∈ ℤ) → (𝑛 − (♯‘𝑇)) ∈ (0..^(♯‘𝑈)))
173	168, 171, 172	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − (♯‘𝑇)) ∈ (0..^(♯‘𝑈)))
174		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝜑)
175	156	eldifad 3917	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}))
176	174, 175	jca 519	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})))
177		eldifi 4085	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
178	177, 159, 161	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ∈ ℂ)
179	178	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ∈ ℂ)
180	71	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ∈ ℂ)
181		eldifsni 4751	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ≠ (♯‘𝑇))
182	181	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ≠ (♯‘𝑇))
183	179, 180, 182	subne0d 11562	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − (♯‘𝑇)) ≠ 0)
184		nelsn 4626	. . . . . . . . . . 11 ⊢ ((𝑛 − (♯‘𝑇)) ≠ 0 → ¬ (𝑛 − (♯‘𝑇)) ∈ {0})
185	176, 183, 184	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ¬ (𝑛 − (♯‘𝑇)) ∈ {0})
186	173, 185	eldifd 3916	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − (♯‘𝑇)) ∈ ((0..^(♯‘𝑈)) ∖ {0}))
187		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑈) = (♯‘𝑈))
188	187, 4	wrdfd 14542	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈:(0..^(♯‘𝑈))⟶𝐴)
189	188	fdmd 6702	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝑈 = (0..^(♯‘𝑈)))
190	189	difeq1d 4080	. . . . . . . . . . 11 ⊢ (𝜑 → (dom 𝑈 ∖ {0}) = ((0..^(♯‘𝑈)) ∖ {0}))
191	190	eleq2d 2849	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑛 − (♯‘𝑇)) ∈ (dom 𝑈 ∖ {0}) ↔ (𝑛 − (♯‘𝑇)) ∈ ((0..^(♯‘𝑈)) ∖ {0})))
192	191	biimpar 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 − (♯‘𝑇)) ∈ ((0..^(♯‘𝑈)) ∖ {0})) → (𝑛 − (♯‘𝑇)) ∈ (dom 𝑈 ∖ {0}))
193	186, 192	syldan 600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − (♯‘𝑇)) ∈ (dom 𝑈 ∖ {0}))
194	167, 193	chnltm1 18651	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑈‘((𝑛 − (♯‘𝑇)) − 1)) < (𝑈‘(𝑛 − (♯‘𝑇))))
195	166, 194	eqbrtrd 5123	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑈‘((𝑛 − 1) − (♯‘𝑇))) < (𝑈‘(𝑛 − (♯‘𝑇))))
196	2	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑇 ∈ Word 𝐴)
197	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → 𝑈 ∈ Word 𝐴)
198	177, 159	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 ∈ ℤ)
199	198	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ∈ ℤ)
200		nn0z 12602	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑇) ∈ ℕ₀ → (♯‘𝑇) ∈ ℤ)
201	2, 29, 200	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑇) ∈ ℤ)
202	201	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ∈ ℤ)
203	199, 202	jca 519	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ))
204		elfzole1 13683	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) → (♯‘𝑇) ≤ 𝑛)
205	177, 204	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → (♯‘𝑇) ≤ 𝑛)
206	205	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ≤ 𝑛)
207		eldifn 4086	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → ¬ 𝑛 ∈ {(♯‘𝑇)})
208		velsn 4599	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ {(♯‘𝑇)} ↔ 𝑛 = (♯‘𝑇))
209	208	biimpri 230	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (♯‘𝑇) → 𝑛 ∈ {(♯‘𝑇)})
210	209	necon3bi 2984	. . . . . . . . . . . . 13 ⊢ (¬ 𝑛 ∈ {(♯‘𝑇)} → 𝑛 ≠ (♯‘𝑇))
211	210	necomd 3013	. . . . . . . . . . . 12 ⊢ (¬ 𝑛 ∈ {(♯‘𝑇)} → (♯‘𝑇) ≠ 𝑛)
212	207, 211	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → (♯‘𝑇) ≠ 𝑛)
213	212	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ≠ 𝑛)
214		simp1r 1213	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ∈ ℤ)
215	214	zred 12687	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ∈ ℝ)
216		simp1l 1212	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 𝑛 ∈ ℤ)
217	216	zred 12687	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 𝑛 ∈ ℝ)
218		simp2 1151	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ≤ 𝑛)
219		simp3 1152	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ≠ 𝑛)
220	219	necomd 3013	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 𝑛 ≠ (♯‘𝑇))
221	215, 217, 218, 220	leneltd 11348	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) < 𝑛)
222		simp1 1150	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ))
223	222	ancomd 465	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) ∈ ℤ ∧ 𝑛 ∈ ℤ))
224		zltp1le 12631	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑇) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((♯‘𝑇) < 𝑛 ↔ ((♯‘𝑇) + 1) ≤ 𝑛))
225	223, 224	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) < 𝑛 ↔ ((♯‘𝑇) + 1) ≤ 𝑛))
226	221, 225	mpbid 234	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) + 1) ≤ 𝑛)
227		peano2re 11367	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℝ → ((♯‘𝑇) + 1) ∈ ℝ)
228	215, 227	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((♯‘𝑇) + 1) ∈ ℝ)
229		1red 11193	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → 1 ∈ ℝ)
230	228, 217, 229	lesub1d 11805	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (((♯‘𝑇) + 1) ≤ 𝑛 ↔ (((♯‘𝑇) + 1) − 1) ≤ (𝑛 − 1)))
231		zcn 12583	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑇) ∈ ℤ → (♯‘𝑇) ∈ ℂ)
232		1cnd 11186	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑇) ∈ ℤ → 1 ∈ ℂ)
233	231, 232	pncand 11554	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℤ → (((♯‘𝑇) + 1) − 1) = (♯‘𝑇))
234	233	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) → (((♯‘𝑇) + 1) − 1) = (♯‘𝑇))
235	234	3ad2ant1 1147	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (((♯‘𝑇) + 1) − 1) = (♯‘𝑇))
236	235	breq1d 5111	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → ((((♯‘𝑇) + 1) − 1) ≤ (𝑛 − 1) ↔ (♯‘𝑇) ≤ (𝑛 − 1)))
237	230, 236	bitrd 281	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (((♯‘𝑇) + 1) ≤ 𝑛 ↔ (♯‘𝑇) ≤ (𝑛 − 1)))
238	226, 237	mpbid 234	. . . . . . . . . 10 ⊢ (((𝑛 ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ) ∧ (♯‘𝑇) ≤ 𝑛 ∧ (♯‘𝑇) ≠ 𝑛) → (♯‘𝑇) ≤ (𝑛 − 1))
239	203, 206, 213, 238	syl3anc 1392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (♯‘𝑇) ≤ (𝑛 − 1))
240	199	zred 12687	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 ∈ ℝ)
241		peano2rem 11509	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℝ → (𝑛 − 1) ∈ ℝ)
242	240, 241	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) ∈ ℝ)
243	201, 170	zaddcld 12691	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)
244	243	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)
245	244	zred 12687	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℝ)
246	240	ltm1d 12134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) < 𝑛)
247		elfzolt2 13684	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) → 𝑛 < ((♯‘𝑇) + (♯‘𝑈)))
248	177, 247	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) → 𝑛 < ((♯‘𝑇) + (♯‘𝑈)))
249	248	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → 𝑛 < ((♯‘𝑇) + (♯‘𝑈)))
250	242, 240, 245, 246, 249	lttrd 11355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) < ((♯‘𝑇) + (♯‘𝑈)))
251		peano2zm 12624	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℤ → (𝑛 − 1) ∈ ℤ)
252	199, 251	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) ∈ ℤ)
253		elfzo 13676	. . . . . . . . . 10 ⊢ (((𝑛 − 1) ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ) → ((𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ↔ ((♯‘𝑇) ≤ (𝑛 − 1) ∧ (𝑛 − 1) < ((♯‘𝑇) + (♯‘𝑈)))))
254	252, 202, 244, 253	syl3anc 1392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → ((𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ↔ ((♯‘𝑇) ≤ (𝑛 − 1) ∧ (𝑛 − 1) < ((♯‘𝑇) + (♯‘𝑈)))))
255	239, 250, 254	mpbir2and 723	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})) → (𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
256	157, 255	sylan2 602	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
257		ccatval2 14601	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑈‘((𝑛 − 1) − (♯‘𝑇))))
258	196, 197, 256, 257	syl3anc 1392	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) = (𝑈‘((𝑛 − 1) − (♯‘𝑇))))
259		ccatval2 14601	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴 ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑈‘(𝑛 − (♯‘𝑇))))
260	196, 197, 168, 259	syl3anc 1392	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘𝑛) = (𝑈‘(𝑛 − (♯‘𝑇))))
261	195, 258, 260	3brtr4d 5133	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
262	261	adantlr 725	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) ∧ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
263		ccatlen 14598	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑈 ∈ Word 𝐴) → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
264	2, 4, 263	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
265	264	eqcomd 2769	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑇) + (♯‘𝑈)) = (♯‘(𝑇 ++ 𝑈)))
266	265, 6	wrdfd 14542	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ++ 𝑈):(0..^((♯‘𝑇) + (♯‘𝑈)))⟶𝐴)
267	266	fdmd 6702	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑇 ++ 𝑈) = (0..^((♯‘𝑇) + (♯‘𝑈))))
268	267	difeq1d 4080	. . . . . . . . 9 ⊢ (𝜑 → (dom (𝑇 ++ 𝑈) ∖ {0}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
269		fzonel 13689	. . . . . . . . . . . 12 ⊢ ¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ (0..^((♯‘𝑇) + (♯‘𝑈)))
270		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
271	270	eldifad 3917	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0})) → ((♯‘𝑇) + (♯‘𝑈)) ∈ (0..^((♯‘𝑇) + (♯‘𝑈))))
272	271	ex 416	. . . . . . . . . . . 12 ⊢ (𝜑 → (((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) → ((♯‘𝑇) + (♯‘𝑈)) ∈ (0..^((♯‘𝑇) + (♯‘𝑈)))))
273	269, 272	mtoi 201	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
274		difsn 4759	. . . . . . . . . . . 12 ⊢ (¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) → (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}))
275	274	eqcomd 2769	. . . . . . . . . . 11 ⊢ (¬ ((♯‘𝑇) + (♯‘𝑈)) ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) → ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) = (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}))
276	273, 275	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) = (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))}))
277		difpr 4764	. . . . . . . . . 10 ⊢ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) = (((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) ∖ {((♯‘𝑇) + (♯‘𝑈))})
278	276, 277	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝜑 → ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
279	268, 278	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → (dom (𝑇 ++ 𝑈) ∖ {0}) = ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
280	279	eleq2d 2849	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0}) ↔ 𝑛 ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
281		eldif 3915	. . . . . . 7 ⊢ (𝑛 ∈ ((0..^((♯‘𝑇) + (♯‘𝑈))) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))}))
282	280, 281	bitrdi 289	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0}) ↔ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})))
283		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → 𝑛 ∈ (0..^(♯‘𝑇)))
284		simplrr 787	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
285	283, 284	eldifd 3916	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → 𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))
286	285	orcd 884	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ (0..^(♯‘𝑇))) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
287		exmidd 906	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} ∨ ¬ 𝑛 ∈ {(♯‘𝑇)}))
288		idd 24	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ {(♯‘𝑇)}))
289		simplrr 787	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})
290	288, 289	jctird 534	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} → (𝑛 ∈ {(♯‘𝑇)} ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})))
291		eldif 3915	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ (𝑛 ∈ {(♯‘𝑇)} ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))}))
292	290, 291	imbitrrdi 254	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
293		idd 24	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → ¬ 𝑛 ∈ {(♯‘𝑇)}))
294		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
295	293, 294	jctild 533	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {(♯‘𝑇)})))
296		eldif 3915	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ↔ (𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {(♯‘𝑇)}))
297	295, 296	imbitrrdi 254	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)})))
298	297, 289	jctird 534	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})))
299		eldif 3915	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ↔ (𝑛 ∈ (((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))}))
300	298, 299	imbitrrdi 254	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (¬ 𝑛 ∈ {(♯‘𝑇)} → 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
301	292, 300	orim12d 977	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ((𝑛 ∈ {(♯‘𝑇)} ∨ ¬ 𝑛 ∈ {(♯‘𝑇)}) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
302	287, 301	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
303	302	olcd 885	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) ∧ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
304	201	anim1ci 625	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈)))) → (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑇) ∈ ℤ))
305	304	adantrr 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑇) ∈ ℤ))
306		fzospliti 13707	. . . . . . . 8 ⊢ ((𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑛 ∈ (0..^(♯‘𝑇)) ∨ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))))
307	305, 306	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 ∈ (0..^(♯‘𝑇)) ∨ 𝑛 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))))
308	286, 303, 307	mpjaodan 971	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (0..^((♯‘𝑇) + (♯‘𝑈))) ∧ ¬ 𝑛 ∈ {0, ((♯‘𝑇) + (♯‘𝑈))})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
309	282, 308	sylbida 601	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
310		3orass 1102	. . . . 5 ⊢ ((𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})) ↔ (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ (𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}))))
311	309, 310	sylibr 236	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) → (𝑛 ∈ ((0..^(♯‘𝑇)) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ({(♯‘𝑇)} ∖ {0, ((♯‘𝑇) + (♯‘𝑈))}) ∨ 𝑛 ∈ ((((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))) ∖ {(♯‘𝑇)}) ∖ {0, ((♯‘𝑇) + (♯‘𝑈))})))
312	40, 155, 262, 311	mpjao3dan 1453	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})) → ((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
313	312	ralrimiva 3155	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛))
314		ischn 18649	. 2 ⊢ ((𝑇 ++ 𝑈) ∈ ( < Chain 𝐴) ↔ ((𝑇 ++ 𝑈) ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom (𝑇 ++ 𝑈) ∖ {0})((𝑇 ++ 𝑈)‘(𝑛 − 1)) < ((𝑇 ++ 𝑈)‘𝑛)))
315	6, 313, 314	sylanbrc 592	1 ⊢ (𝜑 → (𝑇 ++ 𝑈) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∨ w3o 1098 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∀wral 3077 Vcvv 3455 ∖ cdif 3902 ⊆ wss 3905 ∅c0 4286 {csn 4583 {cpr 4585 class class class wbr 5101 dom cdm 5648 ‘cfv 6521 (class class class)co 7396 ℂcc 11082 ℝcr 11083 0cc0 11084 1c1 11085 + caddc 11087 < clt 11227 ≤ cle 11228 − cmin 11425 ℕcn 12220 ℕ₀cn0 12491 ℤcz 12578 ..^cfzo 13669 ♯chash 14353 Word cword 14536 lastSclsw 14585 ++ cconcat 14593 Chain cchn 18647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-n0 12492 df-z 12579 df-uz 12850 df-fz 13523 df-fzo 13670 df-hash 14354 df-word 14537 df-lsw 14586 df-concat 14594 df-chn 18648

This theorem is referenced by: (None)

W3C validator