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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 7153 | . . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵)) | |
2 | 1 | oveq1d 7160 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶)) |
3 | df-dec 12087 | . 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶) | |
4 | df-dec 12087 | . 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶) | |
5 | 2, 3, 4 | 3eqtr4g 2878 | 1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 (class class class)co 7145 1c1 10526 + caddc 10528 · cmul 10530 9c9 11687 ;cdc 12086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-dec 12087 |
This theorem is referenced by: deceq1i 12093 |
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