chnsubseq - Mathbox for Ender Ting

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Theorem chnsubseq 47456

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 47454	. 2 ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ Word 𝐴)
4		chnsubseq.3	. . . . . 6 ⊢ (𝜑 → < Po 𝐴)
5	4	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → < Po 𝐴)
6	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑊 ∈ ( < Chain 𝐴))
7	2	chnwrd 18640	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Word (0..^(♯‘𝑊)))
8	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼 ∈ Word (0..^(♯‘𝑊)))
9		wrdf 14531	. . . . . . 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 4084	. . . . . . 7 ⊢ (𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}) → 𝑥 ∈ dom (𝑊 ∘ 𝐼))
12		wrddm 14534	. . . . . . . . . . 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 47455	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
15	14	oveq2d 7412	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘(𝑊 ∘ 𝐼))) = (0..^(♯‘𝐼)))
16	13, 15	eqtrd 2797	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘𝐼)))
17	16	eleq2d 2848	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ dom (𝑊 ∘ 𝐼) ↔ 𝑥 ∈ (0..^(♯‘𝐼))))
18	17	biimpa 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝑊 ∘ 𝐼)) → 𝑥 ∈ (0..^(♯‘𝐼)))
19	11, 18	sylan2 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘𝐼)))
20	10, 19	ffvelcdmd 7066	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘𝑥) ∈ (0..^(♯‘𝑊)))
21		elfzonn0 13713	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘𝐼)) → 𝑥 ∈ ℕ₀)
22	19, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℕ₀)
23		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}))
24	23	eldifbd 3917	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ¬ 𝑥 ∈ {0})
25		velsn 4598	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
26	24, 25	sylnib 330	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ¬ 𝑥 = 0)
27	26	neqned 2964	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ≠ 0)
28		elnnne0 12495	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↔ (𝑥 ∈ ℕ₀ ∧ 𝑥 ≠ 0))
29	22, 27, 28	sylanbrc 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℕ)
30		nnm1ge0 12641	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (𝑥 − 1))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ≤ (𝑥 − 1))
32	22	nn0red 12543	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℝ)
33		peano2rem 11498	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℝ)
35		lencl 14546	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Word (0..^(♯‘𝑊)) → (♯‘𝐼) ∈ ℕ₀)
36	7, 35	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ₀)
37	36	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℕ₀)
38	37	nn0red 12543	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℝ)
39	32	ltm1d 12124	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) < 𝑥)
40	11	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ dom (𝑊 ∘ 𝐼))
41	13	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → dom (𝑊 ∘ 𝐼) = (0..^(♯‘(𝑊 ∘ 𝐼))))
42	40, 41	eleqtrd 2864	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘(𝑊 ∘ 𝐼))))
43		elfzolt2 13674	. . . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
46	44, 45	breqtrd 5126	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 < (♯‘𝐼))
47	34, 32, 38, 39, 46	lttrd 11344	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) < (♯‘𝐼))
48		elfzoelz 13664	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘𝐼)) → 𝑥 ∈ ℤ)
49	19, 48	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℤ)
50		peano2zm 12614	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (𝑥 − 1) ∈ ℤ)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℤ)
52		0zd 12580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ∈ ℤ)
53	36	nn0zd 12593	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐼) ∈ ℤ)
54	53	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℤ)
55		elfzo 13666	. . . . . . . . . 10 ⊢ (((𝑥 − 1) ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐼) ∈ ℤ) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
56	51, 52, 54, 55	syl3anc 1390	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
57	31, 47, 56	mpbir2and 723	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ (0..^(♯‘𝐼)))
58	10, 57	ffvelcdmd 7066	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(♯‘𝑊)))
59		elfzonn0 13713	. . . . . . 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 13664	. . . . . . 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 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))
64		wrddm 14534	. . . . . . . . . . . 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 2807	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = dom 𝐼)
67	66	difeq1d 4079	. . . . . . . . 9 ⊢ (𝜑 → (dom (𝑊 ∘ 𝐼) ∖ {0}) = (dom 𝐼 ∖ {0}))
68	67	eleq2d 2848	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}) ↔ 𝑥 ∈ (dom 𝐼 ∖ {0})))
69	68	biimpa 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (dom 𝐼 ∖ {0}))
70	63, 69	chnltm1 18641	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) < (𝐼‘𝑥))
71		elfzo0z 13707	. . . . . 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 1357	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(𝐼‘𝑥)))
73	5, 6, 20, 72	chnlt 18655	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑊‘(𝐼‘(𝑥 − 1))) < (𝑊‘(𝐼‘𝑥)))
74	10, 57	fvco3d 6968	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘(𝑥 − 1)) = (𝑊‘(𝐼‘(𝑥 − 1))))
75	10, 19	fvco3d 6968	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘𝑥) = (𝑊‘(𝐼‘𝑥)))
76	73, 74, 75	3brtr4d 5132	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥))
77	76	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥))
78		ischn 18639	. 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 592	1 ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∀wral 3076 ∖ cdif 3901 {csn 4582 class class class wbr 5100 Po wpo 5553 dom cdm 5647 ∘ ccom 5651 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ℝcr 11072 0cc0 11073 1c1 11074 < clt 11216 ≤ cle 11217 − cmin 11414 ℕcn 12210 ℕ₀cn0 12481 ℤcz 12568 ..^cfzo 13659 ♯chash 14343 Word cword 14526 Chain cchn 18637

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 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 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-n0 12482 df-xnn0 12555 df-z 12569 df-uz 12840 df-rp 12994 df-fz 13513 df-fzo 13660 df-hash 14344 df-word 14527 df-lsw 14576 df-concat 14584 df-s1 14610 df-substr 14655 df-pfx 14685 df-chn 18638

This theorem is referenced by: (None)

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