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Mirrors > Home > ILE Home > Th. List > mvrladdd | GIF version |
Description: Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
mvrraddd.1 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
mvrraddd.2 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
mvrraddd.3 | ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) |
Ref | Expression |
---|---|
mvrladdd | ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvrraddd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
2 | mvrraddd.1 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | mvrraddd.3 | . . 3 ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶)) | |
4 | 2, 1, 3 | comraddd 8069 | . 2 ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐵)) |
5 | 1, 2, 4 | mvrraddd 8278 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5851 ℂcc 7765 + caddc 7770 − cmin 8083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7859 ax-1cn 7860 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-sub 8085 |
This theorem is referenced by: dveflem 13446 tangtx 13518 |
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