Theorem chnltm1 32934

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

Hypotheses

Ref	Expression
chnwrd.1	⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))
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 7410	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘(𝑛 − 1)) = (𝐶‘(𝑁 − 1)))
2		fveq2 6858	. . 3 ⊢ (𝑛 = 𝑁 → (𝐶‘𝑛) = (𝐶‘𝑁))
3	1, 2	breq12d 5120	. 2 ⊢ (𝑛 = 𝑁 → ((𝐶‘(𝑛 − 1)) < (𝐶‘𝑛) ↔ (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁)))
4		chnwrd.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ( < Chain𝐴))
5		ischn 32932	. . . 4 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
6	4, 5	sylib 218	. . 3 ⊢ (𝜑 → (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
7	6	simprd 495	. 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 3589	1 ⊢ (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3911  {csn 4589   class class class wbr 5107  dom cdm 5638  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069   − cmin 11405  Word cword 14478  Chaincchn 32930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-dm 5648  df-iota 6464  df-fv 6519  df-ov 7390  df-chn 32931

This theorem is referenced by:  pfxchn  32935