chnsubseq - Mathbox for Ender Ting

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Theorem chnsubseq 47238

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 47236	. 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 18543	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Word (0..^(♯‘𝑊)))
8	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼 ∈ Word (0..^(♯‘𝑊)))
9		wrdf 14453	. . . . . . 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 4085	. . . . . . 7 ⊢ (𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}) → 𝑥 ∈ dom (𝑊 ∘ 𝐼))
12		wrddm 14456	. . . . . . . . . . 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 47237	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
15	14	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘(𝑊 ∘ 𝐼))) = (0..^(♯‘𝐼)))
16	13, 15	eqtrd 2772	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘𝐼)))
17	16	eleq2d 2823	. . . . . . . 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 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘𝐼)))
20	10, 19	ffvelcdmd 7039	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘𝑥) ∈ (0..^(♯‘𝑊)))
21		elfzonn0 13635	. . . . . . . . . . . 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 3916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ¬ 𝑥 ∈ {0})
25		velsn 4598	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {0} ↔ 𝑥 = 0)
26	24, 25	sylnib 328	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ¬ 𝑥 = 0)
27	26	neqned 2940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ≠ 0)
28		elnnne0 12427	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↔ (𝑥 ∈ ℕ₀ ∧ 𝑥 ≠ 0))
29	22, 27, 28	sylanbrc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℕ)
30		nnm1ge0 12572	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (𝑥 − 1))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ≤ (𝑥 − 1))
32	22	nn0red 12475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℝ)
33		peano2rem 11460	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℝ)
35		lencl 14468	. . . . . . . . . . . . 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 12475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℝ)
39	32	ltm1d 12086	. . . . . . . . . 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 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘(𝑊 ∘ 𝐼))))
43		elfzolt2 13596	. . . . . . . . . . . 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 5126	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 < (♯‘𝐼))
47	34, 32, 38, 39, 46	lttrd 11306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) < (♯‘𝐼))
48		elfzoelz 13587	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘𝐼)) → 𝑥 ∈ ℤ)
49	19, 48	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℤ)
50		peano2zm 12546	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (𝑥 − 1) ∈ ℤ)
51	49, 50	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℤ)
52		0zd 12512	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ∈ ℤ)
53	36	nn0zd 12525	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐼) ∈ ℤ)
54	53	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℤ)
55		elfzo 13589	. . . . . . . . . 10 ⊢ (((𝑥 − 1) ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐼) ∈ ℤ) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
56	51, 52, 54, 55	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
57	31, 47, 56	mpbir2and 714	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ (0..^(♯‘𝐼)))
58	10, 57	ffvelcdmd 7039	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(♯‘𝑊)))
59		elfzonn0 13635	. . . . . . 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 13587	. . . . . . 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 14456	. . . . . . . . . . . 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 2782	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = dom 𝐼)
67	66	difeq1d 4079	. . . . . . . . 9 ⊢ (𝜑 → (dom (𝑊 ∘ 𝐼) ∖ {0}) = (dom 𝐼 ∖ {0}))
68	67	eleq2d 2823	. . . . . . . 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 18544	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) < (𝐼‘𝑥))
71		elfzo0z 13629	. . . . . 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 1345	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(𝐼‘𝑥)))
73	5, 6, 20, 72	chnlt 18558	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑊‘(𝐼‘(𝑥 − 1))) < (𝑊‘(𝐼‘𝑥)))
74	10, 57	fvco3d 6942	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘(𝑥 − 1)) = (𝑊‘(𝐼‘(𝑥 − 1))))
75	10, 19	fvco3d 6942	. . . 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 3130	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥))
78		ischn 18542	. 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 584	1 ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∖ cdif 3900 {csn 4582 class class class wbr 5100 Po wpo 5538 dom cdm 5632 ∘ ccom 5636 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 0cc0 11038 1c1 11039 < clt 11178 ≤ cle 11179 − cmin 11376 ℕcn 12157 ℕ₀cn0 12413 ℤcz 12500 ..^cfzo 13582 ♯chash 14265 Word cword 14448 Chain cchn 18540

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

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 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-lsw 14498 df-concat 14506 df-s1 14532 df-substr 14577 df-pfx 14607 df-chn 18541

This theorem is referenced by: (None)

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