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Mirrors > Home > MPE Home > Th. List > deceq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
deceq2 | โข (๐ด = ๐ต โ ;๐ถ๐ด = ;๐ถ๐ต) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7419 | . 2 โข (๐ด = ๐ต โ (((9 + 1) ยท ๐ถ) + ๐ด) = (((9 + 1) ยท ๐ถ) + ๐ต)) | |
2 | df-dec 12682 | . 2 โข ;๐ถ๐ด = (((9 + 1) ยท ๐ถ) + ๐ด) | |
3 | df-dec 12682 | . 2 โข ;๐ถ๐ต = (((9 + 1) ยท ๐ถ) + ๐ต) | |
4 | 1, 2, 3 | 3eqtr4g 2795 | 1 โข (๐ด = ๐ต โ ;๐ถ๐ด = ;๐ถ๐ต) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1539 (class class class)co 7411 1c1 11113 + caddc 11115 ยท cmul 11117 9c9 12278 ;cdc 12681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-ov 7414 df-dec 12682 |
This theorem is referenced by: deceq2i 12689 |
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