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Mirrors > Home > MPE Home > Th. List > s6eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 6 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) |
s5eqd.5 | ⊢ (𝜑 → 𝐸 = 𝑅) |
s6eqd.6 | ⊢ (𝜑 → 𝐹 = 𝑆) |
Ref | Expression |
---|---|
s6eqd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = 〈“𝑁𝑂𝑃𝑄𝑅𝑆”〉) |
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 | s4eqd.4 | . . . 4 ⊢ (𝜑 → 𝐷 = 𝑄) | |
5 | s5eqd.5 | . . . 4 ⊢ (𝜑 → 𝐸 = 𝑅) | |
6 | 1, 2, 3, 4, 5 | s5eqd 14396 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸”〉 = 〈“𝑁𝑂𝑃𝑄𝑅”〉) |
7 | s6eqd.6 | . . . 4 ⊢ (𝜑 → 𝐹 = 𝑆) | |
8 | 7 | s1eqd 14123 | . . 3 ⊢ (𝜑 → 〈“𝐹”〉 = 〈“𝑆”〉) |
9 | 6, 8 | oveq12d 7209 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷𝐸”〉 ++ 〈“𝐹”〉) = (〈“𝑁𝑂𝑃𝑄𝑅”〉 ++ 〈“𝑆”〉)) |
10 | df-s6 14382 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = (〈“𝐴𝐵𝐶𝐷𝐸”〉 ++ 〈“𝐹”〉) | |
11 | df-s6 14382 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄𝑅𝑆”〉 = (〈“𝑁𝑂𝑃𝑄𝑅”〉 ++ 〈“𝑆”〉) | |
12 | 9, 10, 11 | 3eqtr4g 2796 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = 〈“𝑁𝑂𝑃𝑄𝑅𝑆”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 (class class class)co 7191 ++ cconcat 14090 〈“cs1 14117 〈“cs5 14374 〈“cs6 14375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-s1 14118 df-s2 14378 df-s3 14379 df-s4 14380 df-s5 14381 df-s6 14382 |
This theorem is referenced by: s7eqd 14398 |
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