Theorem chnrin 47822

Description: Satisfying two chain relations makes a chain under their intersection. (Contributed by Ender Ting, 24-Jul-2026.)

Assertion

Ref	Expression
chnrin	⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∩ < ) Chain 𝐵))

Proof of Theorem chnrin

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ischn 18728	. . . 4 ⊢ (𝐴 ∈ (𝑅 Chain 𝐵) ↔ (𝐴 ∈ Word 𝐵 ∧ ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1))𝑅(𝐴‘𝑛)))
2	1	simplbi 502	. . 3 ⊢ (𝐴 ∈ (𝑅 Chain 𝐵) → 𝐴 ∈ Word 𝐵)
3	2	adantr 486	. 2 ⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ Word 𝐵)
4	1	simprbi 503	. . . . . 6 ⊢ (𝐴 ∈ (𝑅 Chain 𝐵) → ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1))𝑅(𝐴‘𝑛))
5	4	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1))𝑅(𝐴‘𝑛))
6	5	r19.21bi 3254	. . . 4 ⊢ (((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) ∧ 𝑛 ∈ (dom 𝐴 ∖ {0})) → (𝐴‘(𝑛 − 1))𝑅(𝐴‘𝑛))
7		ischn 18728	. . . . . . 7 ⊢ (𝐴 ∈ ( < Chain 𝐵) ↔ (𝐴 ∈ Word 𝐵 ∧ ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1)) < (𝐴‘𝑛)))
8	7	simprbi 503	. . . . . 6 ⊢ (𝐴 ∈ ( < Chain 𝐵) → ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1)) < (𝐴‘𝑛))
9	8	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1)) < (𝐴‘𝑛))
10	9	r19.21bi 3254	. . . 4 ⊢ (((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) ∧ 𝑛 ∈ (dom 𝐴 ∖ {0})) → (𝐴‘(𝑛 − 1)) < (𝐴‘𝑛))
11		brin 5157	. . . 4 ⊢ ((𝐴‘(𝑛 − 1))(𝑅 ∩ < )(𝐴‘𝑛) ↔ ((𝐴‘(𝑛 − 1))𝑅(𝐴‘𝑛) ∧ (𝐴‘(𝑛 − 1)) < (𝐴‘𝑛)))
12	6, 10, 11	sylanbrc 595	. . 3 ⊢ (((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) ∧ 𝑛 ∈ (dom 𝐴 ∖ {0})) → (𝐴‘(𝑛 − 1))(𝑅 ∩ < )(𝐴‘𝑛))
13	12	ralrimiva 3154	. 2 ⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1))(𝑅 ∩ < )(𝐴‘𝑛))
14		ischn 18728	. 2 ⊢ (𝐴 ∈ ((𝑅 ∩ < ) Chain 𝐵) ↔ (𝐴 ∈ Word 𝐵 ∧ ∀𝑛 ∈ (dom 𝐴 ∖ {0})(𝐴‘(𝑛 − 1))(𝑅 ∩ < )(𝐴‘𝑛)))
15	3, 13, 14	sylanbrc 595	1 ⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∩ < ) Chain 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3896   ∩ cin 3898  {csn 4584   class class class wbr 5103  dom cdm 5648  ‘cfv 6528  (class class class)co 7409  0cc0 11157  1c1 11158   − cmin 11498  Word cword 14611   Chain cchn 18726

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5658  df-iota 6484  df-fv 6536  df-chn 18727

This theorem is used by:  chnrin2  47824