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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 7165 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 + (𝐶 / ;10)) = (𝐵 + (𝐶 / ;10))) | |
2 | df-dp2 30550 | . 2 ⊢ _𝐴𝐶 = (𝐴 + (𝐶 / ;10)) | |
3 | df-dp2 30550 | . 2 ⊢ _𝐵𝐶 = (𝐵 + (𝐶 / ;10)) | |
4 | 1, 2, 3 | 3eqtr4g 2883 | 1 ⊢ (𝐴 = 𝐵 → _𝐴𝐶 = _𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 / cdiv 11299 ;cdc 12101 _cdp2 30549 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-dp2 30550 |
This theorem is referenced by: dp2eq1i 30553 |
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