Theorem chnltm1 18641

Description: Basic property of a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)

Hypotheses

Ref	Expression
chnwrd.1	⊢ (𝜑 → 𝐶 ∈ ( < Chain 𝐴))
chnltm1.2	⊢ (𝜑 → 𝑁 ∈ (dom 𝐶 ∖ {0}))

Assertion

Ref	Expression
chnltm1	⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))

Proof of Theorem chnltm1

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 7419	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘(𝑛 − 1)) = (𝐶‘(𝑁 − 1)))
2		fveq2 6867	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘𝑛) = (𝐶‘𝑁))
3	1, 2	breq12d 5113	. 2 ⊢ (𝑛 = 𝑁 → ((𝐶‘(𝑛 − 1)) < (𝐶‘𝑛) ↔ (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁)))
4		chnwrd.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ( < Chain 𝐴))
5		ischn 18639	. . . 4 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
6	4, 5	sylib 220	. . 3 ⊢ (𝜑 → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
7	6	simprd 499	. 2 ⊢ (𝜑 → ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))
8		chnltm1.2	. 2 ⊢ (𝜑 → 𝑁 ∈ (dom 𝐶 ∖ {0}))
9	3, 7, 8	rspcdva 3582	1 ⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∖ cdif 3901  {csn 4582   class class class wbr 5100  dom cdm 5647  ‘cfv 6521  (class class class)co 7396  0cc0 11073  1c1 11074   − cmin 11414  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-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-dm 5657  df-iota 6477  df-fv 6529  df-ov 7399  df-chn 18638

This theorem is referenced by:  pfxchn  18642  chnccat  18658  chnrev  18659  chnsubseq  47456