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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ischn | Structured version Visualization version GIF version |
Description: Property of being a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.) |
Ref | Expression |
---|---|
ischn | ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5911 | . . . 4 ⊢ (𝑐 = 𝐶 → dom 𝑐 = dom 𝐶) | |
2 | 1 | difeq1d 4135 | . . 3 ⊢ (𝑐 = 𝐶 → (dom 𝑐 ∖ {0}) = (dom 𝐶 ∖ {0})) |
3 | fveq1 6900 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘(𝑛 − 1)) = (𝐶‘(𝑛 − 1))) | |
4 | fveq1 6900 | . . . 4 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑛) = (𝐶‘𝑛)) | |
5 | 3, 4 | breq12d 5162 | . . 3 ⊢ (𝑐 = 𝐶 → ((𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ (𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
6 | 2, 5 | raleqbidv 3342 | . 2 ⊢ (𝑐 = 𝐶 → (∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛) ↔ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
7 | df-chn 32956 | . 2 ⊢ ( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)} | |
8 | 6, 7 | elrab2 3698 | 1 ⊢ (𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ∀wral 3057 ∖ cdif 3960 {csn 4630 class class class wbr 5149 dom cdm 5683 ‘cfv 6558 (class class class)co 7425 0cc0 11146 1c1 11147 − cmin 11483 Word cword 14538 Chaincchn 32955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-dm 5693 df-iota 6510 df-fv 6566 df-chn 32956 |
This theorem is referenced by: chnwrd 32958 chnltm1 32959 pfxchn 32960 chnind 32961 chnso 32964 |
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