![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > s4eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 4 word. (Contributed by Mario Carneiro, 27-Feb-2016.) |
Ref | Expression |
---|---|
s2eqd.1 | ⊢ (𝜑 → 𝐴 = 𝑁) |
s2eqd.2 | ⊢ (𝜑 → 𝐵 = 𝑂) |
s3eqd.3 | ⊢ (𝜑 → 𝐶 = 𝑃) |
s4eqd.4 | ⊢ (𝜑 → 𝐷 = 𝑄) |
Ref | Expression |
---|---|
s4eqd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s2eqd.1 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝑁) | |
2 | s2eqd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = 𝑂) | |
3 | s3eqd.3 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝑃) | |
4 | 1, 2, 3 | s3eqd 14087 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 = 〈“𝑁𝑂𝑃”〉) |
5 | s4eqd.4 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝑄) | |
6 | 5 | s1eqd 13763 | . . 3 ⊢ (𝜑 → 〈“𝐷”〉 = 〈“𝑄”〉) |
7 | 4, 6 | oveq12d 6993 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉)) |
8 | df-s4 14073 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) | |
9 | df-s4 14073 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄”〉 = (〈“𝑁𝑂𝑃”〉 ++ 〈“𝑄”〉) | |
10 | 7, 8, 9 | 3eqtr4g 2834 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷”〉 = 〈“𝑁𝑂𝑃𝑄”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 (class class class)co 6975 ++ cconcat 13732 〈“cs1 13757 〈“cs3 14065 〈“cs4 14066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-rex 3089 df-rab 3092 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-iota 6150 df-fv 6194 df-ov 6978 df-s1 13758 df-s2 14071 df-s3 14072 df-s4 14073 |
This theorem is referenced by: s5eqd 14089 |
Copyright terms: Public domain | W3C validator |