Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7283 | . . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵)) | |
2 | 1 | oveq1d 7290 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶)) |
3 | df-dec 12438 | . 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶) | |
4 | df-dec 12438 | . 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶) | |
5 | 2, 3, 4 | 3eqtr4g 2803 | 1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 (class class class)co 7275 1c1 10872 + caddc 10874 · cmul 10876 9c9 12035 ;cdc 12437 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-dec 12438 |
This theorem is referenced by: deceq1i 12444 |
Copyright terms: Public domain | W3C validator |