Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2eq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
dp2eq2 | ⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7163 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 / ;10) = (𝐵 / ;10)) | |
2 | 1 | oveq2d 7172 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 + (𝐴 / ;10)) = (𝐶 + (𝐵 / ;10))) |
3 | df-dp2 30548 | . 2 ⊢ _𝐶𝐴 = (𝐶 + (𝐴 / ;10)) | |
4 | df-dp2 30548 | . 2 ⊢ _𝐶𝐵 = (𝐶 + (𝐵 / ;10)) | |
5 | 2, 3, 4 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → _𝐶𝐴 = _𝐶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 / cdiv 11297 ;cdc 12099 _cdp2 30547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-dp2 30548 |
This theorem is referenced by: dp2eq2i 30552 |
Copyright terms: Public domain | W3C validator |