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Mirrors > Home > MPE Home > Th. List > deceq1 | 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 |
---|---|
deceq1 | โข (๐ด = ๐ต โ ;๐ด๐ถ = ;๐ต๐ถ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7366 | . . 3 โข (๐ด = ๐ต โ ((9 + 1) ยท ๐ด) = ((9 + 1) ยท ๐ต)) | |
2 | 1 | oveq1d 7373 | . 2 โข (๐ด = ๐ต โ (((9 + 1) ยท ๐ด) + ๐ถ) = (((9 + 1) ยท ๐ต) + ๐ถ)) |
3 | df-dec 12620 | . 2 โข ;๐ด๐ถ = (((9 + 1) ยท ๐ด) + ๐ถ) | |
4 | df-dec 12620 | . 2 โข ;๐ต๐ถ = (((9 + 1) ยท ๐ต) + ๐ถ) | |
5 | 2, 3, 4 | 3eqtr4g 2802 | 1 โข (๐ด = ๐ต โ ;๐ด๐ถ = ;๐ต๐ถ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 (class class class)co 7358 1c1 11053 + caddc 11055 ยท cmul 11057 9c9 12216 ;cdc 12619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-dec 12620 |
This theorem is referenced by: deceq1i 12626 |
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