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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 7167 | . . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵)) | |
2 | 1 | oveq1d 7174 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶)) |
3 | df-dec 12102 | . 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶) | |
4 | df-dec 12102 | . 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶) | |
5 | 2, 3, 4 | 3eqtr4g 2884 | 1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 (class class class)co 7159 1c1 10541 + caddc 10543 · cmul 10545 9c9 11702 ;cdc 12101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-dec 12102 |
This theorem is referenced by: deceq1i 12108 |
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