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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 7416 | . . 3 โข (๐ด = ๐ต โ ((9 + 1) ยท ๐ด) = ((9 + 1) ยท ๐ต)) | |
2 | 1 | oveq1d 7423 | . 2 โข (๐ด = ๐ต โ (((9 + 1) ยท ๐ด) + ๐ถ) = (((9 + 1) ยท ๐ต) + ๐ถ)) |
3 | df-dec 12677 | . 2 โข ;๐ด๐ถ = (((9 + 1) ยท ๐ด) + ๐ถ) | |
4 | df-dec 12677 | . 2 โข ;๐ต๐ถ = (((9 + 1) ยท ๐ต) + ๐ถ) | |
5 | 2, 3, 4 | 3eqtr4g 2797 | 1 โข (๐ด = ๐ต โ ;๐ด๐ถ = ;๐ต๐ถ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 (class class class)co 7408 1c1 11110 + caddc 11112 ยท cmul 11114 9c9 12273 ;cdc 12676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7411 df-dec 12677 |
This theorem is referenced by: deceq1i 12683 |
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