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Theorem chnsubseq 48163

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 48161	. 2 ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ Word 𝐴)
4		chnsubseq.3	. . . . . 6 ⊢ (𝜑 → < Po 𝐴)
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → < Po 𝐴)
6	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑊 ∈ ( < Chain 𝐴))
7	2	chnwrd 18832	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ Word (0..^(♯‘𝑊)))
8	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼 ∈ Word (0..^(♯‘𝑊)))
9		wrdf 14704	. . . . . . 7 ⊢ (𝐼 ∈ Word (0..^(♯‘𝑊)) → 𝐼:(0..^(♯‘𝐼))⟶(0..^(♯‘𝑊)))
10	8, 9	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼:(0..^(♯‘𝐼))⟶(0..^(♯‘𝑊)))
11		eldifi 4070	. . . . . . 7 ⊢ (𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}) → 𝑥 ∈ dom (𝑊 ∘ 𝐼))
12		wrddm 14707	. . . . . . . . . . 11 ⊢ ((𝑊 ∘ 𝐼) ∈ Word 𝐴 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘(𝑊 ∘ 𝐼))))
13	3, 12	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘(𝑊 ∘ 𝐼))))
14	1, 2	chnsubseqwl 48162	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
15	14	oveq2d 7467	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘(𝑊 ∘ 𝐼))) = (0..^(♯‘𝐼)))
16	13, 15	eqtrd 2798	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = (0..^(♯‘𝐼)))
17	16	eleq2d 2850	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ dom (𝑊 ∘ 𝐼) ↔ 𝑥 ∈ (0..^(♯‘𝐼))))
18	17	biimpa 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝑊 ∘ 𝐼)) → 𝑥 ∈ (0..^(♯‘𝐼)))
19	11, 18	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘𝐼)))
20	10, 19	ffvelcdmd 7117	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘𝑥) ∈ (0..^(♯‘𝑊)))
21		elfzonn0 13882	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘𝐼)) → 𝑥 ∈ ℕ₀)
22	19, 21	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℕ₀)
23		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}))
24	23	eldifsnbd 4751	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ≠ 0)
25		elnnne0 12660	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ ↔ (𝑥 ∈ ℕ₀ ∧ 𝑥 ≠ 0))
26	22, 24, 25	sylanbrc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℕ)
27		nnm1ge0 12807	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (𝑥 − 1))
28	26, 27	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ≤ (𝑥 − 1))
29	22	nn0red 12708	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ ℝ)
30		peano2rem 11664	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑥 − 1) ∈ ℝ)
31	29, 30	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℝ)
32		lencl 14719	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ Word (0..^(♯‘𝑊)) → (♯‘𝐼) ∈ ℕ₀)
33	7, 32	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐼) ∈ ℕ₀)
34	33	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℕ₀)
35	34	nn0red 12708	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℝ)
36	29	ltm1d 12289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) < 𝑥)
37	11	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ dom (𝑊 ∘ 𝐼))
38	13	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → dom (𝑊 ∘ 𝐼) = (0..^(♯‘(𝑊 ∘ 𝐼))))
39	37, 38	eleqtrd 2866	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (0..^(♯‘(𝑊 ∘ 𝐼))))
40		elfzolt2 13843	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^(♯‘(𝑊 ∘ 𝐼))) → 𝑥 < (♯‘(𝑊 ∘ 𝐼)))
41	39, 40	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 < (♯‘(𝑊 ∘ 𝐼)))
42	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
43	41, 42	breqtrd 5137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 < (♯‘𝐼))
44	31, 29, 35, 36, 43	lttrd 11510	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) < (♯‘𝐼))
45		elfzoelz 13833	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(♯‘𝐼)) → 𝑥 ∈ ℤ)
46		peano2zm 12779	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℤ → (𝑥 − 1) ∈ ℤ)
47	19, 45, 46	3syl 19	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ ℤ)
48		0zd 12745	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 0 ∈ ℤ)
49	33	nn0zd 12758	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐼) ∈ ℤ)
50	49	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (♯‘𝐼) ∈ ℤ)
51		elfzo 13835	. . . . . . . . . 10 ⊢ (((𝑥 − 1) ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐼) ∈ ℤ) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
52	47, 48, 50, 51	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑥 − 1) ∈ (0..^(♯‘𝐼)) ↔ (0 ≤ (𝑥 − 1) ∧ (𝑥 − 1) < (♯‘𝐼))))
53	28, 44, 52	mpbir2and 726	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑥 − 1) ∈ (0..^(♯‘𝐼)))
54	10, 53	ffvelcdmd 7117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(♯‘𝑊)))
55		elfzonn0 13882	. . . . . . 7 ⊢ ((𝐼‘(𝑥 − 1)) ∈ (0..^(♯‘𝑊)) → (𝐼‘(𝑥 − 1)) ∈ ℕ₀)
56	54, 55	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ ℕ₀)
57		elfzoelz 13833	. . . . . . 7 ⊢ ((𝐼‘𝑥) ∈ (0..^(♯‘𝑊)) → (𝐼‘𝑥) ∈ ℤ)
58	20, 57	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘𝑥) ∈ ℤ)
59	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))
60		wrddm 14707	. . . . . . . . . . . 12 ⊢ (𝐼 ∈ Word (0..^(♯‘𝑊)) → dom 𝐼 = (0..^(♯‘𝐼)))
61	7, 60	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐼 = (0..^(♯‘𝐼)))
62	15, 13, 61	3eqtr4d 2808	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑊 ∘ 𝐼) = dom 𝐼)
63	62	difeq1d 4065	. . . . . . . . 9 ⊢ (𝜑 → (dom (𝑊 ∘ 𝐼) ∖ {0}) = (dom 𝐼 ∖ {0}))
64	63	eleq2d 2850	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0}) ↔ 𝑥 ∈ (dom 𝐼 ∖ {0})))
65	64	biimpa 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → 𝑥 ∈ (dom 𝐼 ∖ {0}))
66	59, 65	chnltm1 18833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) < (𝐼‘𝑥))
67		elfzo0z 13876	. . . . . 6 ⊢ ((𝐼‘(𝑥 − 1)) ∈ (0..^(𝐼‘𝑥)) ↔ ((𝐼‘(𝑥 − 1)) ∈ ℕ₀ ∧ (𝐼‘𝑥) ∈ ℤ ∧ (𝐼‘(𝑥 − 1)) < (𝐼‘𝑥)))
68	56, 58, 66, 67	syl3anbrc 1362	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝐼‘(𝑥 − 1)) ∈ (0..^(𝐼‘𝑥)))
69	5, 6, 20, 68	chnlt 18847	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → (𝑊‘(𝐼‘(𝑥 − 1))) < (𝑊‘(𝐼‘𝑥)))
70	10, 53	fvco3d 7017	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘(𝑥 − 1)) = (𝑊‘(𝐼‘(𝑥 − 1))))
71	10, 19	fvco3d 7017	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘𝑥) = (𝑊‘(𝐼‘𝑥)))
72	69, 70, 71	3brtr4d 5143	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})) → ((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥))
73	72	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥))
74		ischn 18831	. 2 ⊢ ((𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴) ↔ ((𝑊 ∘ 𝐼) ∈ Word 𝐴 ∧ ∀𝑥 ∈ (dom (𝑊 ∘ 𝐼) ∖ {0})((𝑊 ∘ 𝐼)‘(𝑥 − 1)) < ((𝑊 ∘ 𝐼)‘𝑥)))
75	3, 73, 74	sylanbrc 595	1 ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∖ cdif 4048 {csn 4588 class class class wbr 5108 Po wpo 5564 dom cdm 5658 ∘ ccom 5662 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℝcr 11238 0cc0 11239 1c1 11240 < clt 11382 ≤ cle 11383 − cmin 11580 ℕcn 12375 ℕ₀cn0 12646 ℤcz 12733 ..^cfzo 13828 ♯chash 14515 Word cword 14699 Chain cchn 18829

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-rp 13161 df-fz 13680 df-fzo 13829 df-hash 14516 df-word 14700 df-lsw 14749 df-concat 14757 df-s1 14784 df-substr 14831 df-pfx 14863 df-chn 18830

This theorem is used by: (None)

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