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Mirrors > Home > MPE Home > Th. List > s8eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 8 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) |
s7eqd.6 | ⊢ (𝜑 → 𝐺 = 𝑇) |
s8eqd.6 | ⊢ (𝜑 → 𝐻 = 𝑈) |
Ref | Expression |
---|---|
s8eqd | ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩) |
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 | s6eqd.6 | . . . 4 ⊢ (𝜑 → 𝐹 = 𝑆) | |
7 | s7eqd.6 | . . . 4 ⊢ (𝜑 → 𝐺 = 𝑇) | |
8 | 1, 2, 3, 4, 5, 6, 7 | s7eqd 14817 | . . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩) |
9 | s8eqd.6 | . . . 4 ⊢ (𝜑 → 𝐻 = 𝑈) | |
10 | 9 | s1eqd 14549 | . . 3 ⊢ (𝜑 → ⟨“𝐻”⟩ = ⟨“𝑈”⟩) |
11 | 8, 10 | oveq12d 7420 | . 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩)) |
12 | df-s8 14803 | . 2 ⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩) | |
13 | df-s8 14803 | . 2 ⊢ ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩) | |
14 | 11, 12, 13 | 3eqtr4g 2789 | 1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 (class class class)co 7402 ++ cconcat 14518 ⟨“cs1 14543 ⟨“cs7 14795 ⟨“cs8 14796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-ov 7405 df-s1 14544 df-s2 14797 df-s3 14798 df-s4 14799 df-s5 14800 df-s6 14801 df-s7 14802 df-s8 14803 |
This theorem is referenced by: (None) |
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