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Mirrors > Home > MPE Home > Th. List > deceq2 | 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 |
---|---|
deceq2 | ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7414 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵)) | |
2 | df-dec 12675 | . 2 ⊢ ;𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴) | |
3 | df-dec 12675 | . 2 ⊢ ;𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2798 | 1 ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 (class class class)co 7406 1c1 11108 + caddc 11110 · cmul 11112 9c9 12271 ;cdc 12674 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 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 6493 df-fv 6549 df-ov 7409 df-dec 12675 |
This theorem is referenced by: deceq2i 12682 |
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