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Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2eq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the decimal expansion constructor. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
dp2eq1 | ⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7220 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 + (𝐶 / ;10)) = (𝐵 + (𝐶 / ;10))) | |
2 | df-dp2 30866 | . 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10)) | |
3 | df-dp2 30866 | . 2 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10)) | |
4 | 1, 2, 3 | 3eqtr4g 2803 | 1 ⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 (class class class)co 7213 0cc0 10729 1c1 10730 + caddc 10732 / cdiv 11489 ;cdc 12293 _cdp2 30865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-ov 7216 df-dp2 30866 |
This theorem is referenced by: dp2eq1i 30869 |
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