chnsubseq - Mathbox for Ender Ting

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Theorem chnsubseq 46988

Description: An order-preserving subsequence of an ordered chain is itself a chain. (Contributed by Ender Ting, 22-Jan-2026.)

**Hypotheses**
Ref	Expression
chnsubseq.1	⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴))
chnsubseq.2	⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))
chnsubseq.3	⊢ (𝜑 → < Po 𝐴)

**Assertion**
Ref	Expression
chnsubseq	⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))

Proof of Theorem chnsubseq Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chnsubseq.1	. . 3 ⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴))
2		chnsubseq.2	. . 3 ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))
3	1, 2	chnsubseqword 46986	. 2 ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ Word 𝐴)
4		chnsubseq.3	. . . . . 6 ⊢ (𝜑 → < Po 𝐴)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → < Po 𝐴)
6	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑊 ∈ ( < Chain 𝐴))
7	2	chnwrd 18514	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Word (0..^(♯‘𝑊)))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼 ∈ Word (0..^(♯‘𝑊)))
9		wrdf 14425	. . . . . . 7 ⊢ (𝐼 ∈ Word (0..^(♯‘𝑊)) → 𝐼:(0..^(♯‘𝐼))⟶(0..^(♯‘𝑊)))
10	8, 9	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼:(0..^(♯‘𝐼))⟶(0..^(♯‘𝑊)))
11		eldifi 4078	. . . . . . 7 ⊢ (𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}) → 𝑥 ∈ dom (𝑊 ∘ 𝐼))
12		wrddm 14428	. . . . . . . . . . 11 ⊢ ((𝑊 ∘ 𝐼) ∈ Word 𝐴 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘(𝑊 ∘ 𝐼))))
13	3, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘(𝑊 ∘ 𝐼))))
14	1, 2	chnsubseqwl 46987	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
15	14	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘(𝑊 ∘ 𝐼))) = (0..^(♯‘𝐼)))
16	13, 15	eqtrd 2766	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘𝐼)))
17	16	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ dom (𝑊 ∘ 𝐼) ↔ 𝑥 ∈ (0..^(♯‘𝐼))))
18	17	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝑊 ∘ 𝐼)) → 𝑥 ∈ (0..^(♯‘𝐼)))
19	11, 18	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘𝐼)))
20	10, 19	ffvelcdmd 7018	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘𝑥) ∈ (0..^(♯‘𝑊)))
21		elfzonn0 13607	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘𝐼)) → 𝑥 ∈ ℕ₀)
22	19, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℕ₀)
23		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}))
24	23	eldifbd 3910	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ¬ 𝑥 ∈ {0})
25		velsn 4589	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
26	24, 25	sylnib 328	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ¬ 𝑥 = 0)
27	26	neqned 2935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ≠ 0)
28		elnnne0 12395	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↔ (𝑥 ∈ ℕ₀ ∧ 𝑥 ≠ 0))
29	22, 27, 28	sylanbrc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℕ)
30		nnm1ge0 12541	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (𝑥 − 1))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ≤ (𝑥 − 1))
32	22	nn0red 12443	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℝ)
33		peano2rem 11428	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℝ)
35		lencl 14440	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Word (0..^(♯‘𝑊)) → (♯‘𝐼) ∈ ℕ₀)
36	7, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ₀)
37	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℕ₀)
38	37	nn0red 12443	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℝ)
39	32	ltm1d 12054	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) < 𝑥)
40	11	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ dom (𝑊 ∘ 𝐼))
41	13	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → dom (𝑊 ∘ 𝐼) = (0..^(♯‘(𝑊 ∘ 𝐼))))
42	40, 41	eleqtrd 2833	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘(𝑊 ∘ 𝐼))))
43		elfzolt2 13568	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘(𝑊 ∘ 𝐼))) → 𝑥 < (♯‘(𝑊 ∘ 𝐼)))
44	42, 43	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 < (♯‘(𝑊 ∘ 𝐼)))
45	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
46	44, 45	breqtrd 5115	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 < (♯‘𝐼))
47	34, 32, 38, 39, 46	lttrd 11274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) < (♯‘𝐼))
48		elfzoelz 13559	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘𝐼)) → 𝑥 ∈ ℤ)
49	19, 48	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℤ)
50		peano2zm 12515	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (𝑥 − 1) ∈ ℤ)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℤ)
52		0zd 12480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ∈ ℤ)
53	36	nn0zd 12494	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐼) ∈ ℤ)
54	53	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℤ)
55		elfzo 13561	. . . . . . . . . 10 ⊢ (((𝑥 − 1) ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐼) ∈ ℤ) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
56	51, 52, 54, 55	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
57	31, 47, 56	mpbir2and 713	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ (0..^(♯‘𝐼)))
58	10, 57	ffvelcdmd 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(♯‘𝑊)))
59		elfzonn0 13607	. . . . . . 7 ⊢ ((𝐼‘(𝑥 − 1)) ∈ (0..^(♯‘𝑊)) → (𝐼‘(𝑥 − 1)) ∈ ℕ₀)
60	58, 59	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ ℕ₀)
61		elfzoelz 13559	. . . . . . 7 ⊢ ((𝐼‘𝑥) ∈ (0..^(♯‘𝑊)) → (𝐼‘𝑥) ∈ ℤ)
62	20, 61	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘𝑥) ∈ ℤ)
63	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))
64		wrddm 14428	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ Word (0..^(♯‘𝑊)) → dom 𝐼 = (0..^(♯‘𝐼)))
65	7, 64	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐼 = (0..^(♯‘𝐼)))
66	15, 13, 65	3eqtr4d 2776	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = dom 𝐼)
67	66	difeq1d 4072	. . . . . . . . 9 ⊢ (𝜑 → (dom (𝑊 ∘ 𝐼) ∖ {0}) = (dom 𝐼 ∖ {0}))
68	67	eleq2d 2817	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}) ↔ 𝑥 ∈ (dom 𝐼 ∖ {0})))
69	68	biimpa 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (dom 𝐼 ∖ {0}))
70	63, 69	chnltm1 18515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) < (𝐼‘𝑥))
71		elfzo0z 13601	. . . . . 6 ⊢ ((𝐼‘(𝑥 − 1)) ∈ (0..^(𝐼‘𝑥)) ↔ ((𝐼‘(𝑥 − 1)) ∈ ℕ₀ ∧ (𝐼‘𝑥) ∈ ℤ ∧ (𝐼‘(𝑥 − 1)) < (𝐼‘𝑥)))
72	60, 62, 70, 71	syl3anbrc 1344	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(𝐼‘𝑥)))
73	5, 6, 20, 72	chnlt 18529	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑊‘(𝐼‘(𝑥 − 1))) < (𝑊‘(𝐼‘𝑥)))
74	10, 57	fvco3d 6922	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘(𝑥 − 1)) = (𝑊‘(𝐼‘(𝑥 − 1))))
75	10, 19	fvco3d 6922	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘𝑥) = (𝑊‘(𝐼‘𝑥)))
76	73, 74, 75	3brtr4d 5121	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥))
77	76	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥))
78		ischn 18513	. 2 ⊢ ((𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴) ↔ ((𝑊 ∘ 𝐼) ∈ Word 𝐴 ∧ ∀𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥)))
79	3, 77, 78	sylanbrc 583	1 ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∖ cdif 3894 {csn 4573 class class class wbr 5089 Po wpo 5520 dom cdm 5614 ∘ ccom 5618 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 0cc0 11006 1c1 11007 < clt 11146 ≤ cle 11147 − cmin 11344 ℕcn 12125 ℕ₀cn0 12381 ℤcz 12468 ..^cfzo 13554 ♯chash 14237 Word cword 14420 Chain cchn 18511

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-n0 12382 df-xnn0 12455 df-z 12469 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14504 df-substr 14549 df-pfx 14579 df-chn 18512

This theorem is referenced by: (None)

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