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Mirrors > Home > MPE Home > Th. List > s7eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a length 7 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) |
Ref | Expression |
---|---|
s7eqd | ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = 〈“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”〉) |
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 | 1, 2, 3, 4, 5, 6 | s6eqd 14871 | . . 3 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = 〈“𝑁𝑂𝑃𝑄𝑅𝑆”〉) |
8 | s7eqd.6 | . . . 4 ⊢ (𝜑 → 𝐺 = 𝑇) | |
9 | 8 | s1eqd 14604 | . . 3 ⊢ (𝜑 → 〈“𝐺”〉 = 〈“𝑇”〉) |
10 | 7, 9 | oveq12d 7434 | . 2 ⊢ (𝜑 → (〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 ++ 〈“𝐺”〉) = (〈“𝑁𝑂𝑃𝑄𝑅𝑆”〉 ++ 〈“𝑇”〉)) |
11 | df-s7 14857 | . 2 ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 ++ 〈“𝐺”〉) | |
12 | df-s7 14857 | . 2 ⊢ 〈“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”〉 = (〈“𝑁𝑂𝑃𝑄𝑅𝑆”〉 ++ 〈“𝑇”〉) | |
13 | 10, 11, 12 | 3eqtr4g 2791 | 1 ⊢ (𝜑 → 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = 〈“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 (class class class)co 7416 ++ cconcat 14573 〈“cs1 14598 〈“cs6 14849 〈“cs7 14850 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-iota 6498 df-fv 6554 df-ov 7419 df-s1 14599 df-s2 14852 df-s3 14853 df-s4 14854 df-s5 14855 df-s6 14856 df-s7 14857 |
This theorem is referenced by: s8eqd 14873 |
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