Theorem chnltm1 18566

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 7379	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘(𝑛 − 1)) = (𝐶‘(𝑁 − 1)))
2		fveq2 6827	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘𝑛) = (𝐶‘𝑁))
3	1, 2	breq12d 5085	. 2 ⊢ (𝑛 = 𝑁 → ((𝐶‘(𝑛 − 1)) < (𝐶‘𝑛) ↔ (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁)))
4		chnwrd.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ( < Chain 𝐴))
5		ischn 18564	. . . 4 ⊢ (𝐶 ∈ ( < Chain 𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
6	4, 5	sylib 219	. . 3 ⊢ (𝜑 → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
7	6	simprd 496	. 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 3561	1 ⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∖ cdif 3880  {csn 4555   class class class wbr 5072  dom cdm 5618  ‘cfv 6485  (class class class)co 7356  0cc0 11029  1c1 11030   − cmin 11368  Word cword 14466   Chain cchn 18562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-dm 5628  df-iota 6441  df-fv 6493  df-ov 7359  df-chn 18563

This theorem is referenced by:  pfxchn  18567  chnccat  18583  chnrev  18584  chnsubseq  47325