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Mirrors > Home > ILE Home > Th. List > mvlladdi | GIF version |
Description: Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
mvlladdi.1 | ⊢ 𝐴 ∈ ℂ |
mvlladdi.2 | ⊢ 𝐵 ∈ ℂ |
mvlladdi.3 | ⊢ (𝐴 + 𝐵) = 𝐶 |
Ref | Expression |
---|---|
mvlladdi | ⊢ 𝐵 = (𝐶 − 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvlladdi.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
2 | mvlladdi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | 1, 2 | pncan3oi 8169 | . 2 ⊢ ((𝐵 + 𝐴) − 𝐴) = 𝐵 |
4 | 2, 1 | addcomi 8097 | . . . 4 ⊢ (𝐴 + 𝐵) = (𝐵 + 𝐴) |
5 | mvlladdi.3 | . . . 4 ⊢ (𝐴 + 𝐵) = 𝐶 | |
6 | 4, 5 | eqtr3i 2200 | . . 3 ⊢ (𝐵 + 𝐴) = 𝐶 |
7 | 6 | oveq1i 5882 | . 2 ⊢ ((𝐵 + 𝐴) − 𝐴) = (𝐶 − 𝐴) |
8 | 3, 7 | eqtr3i 2200 | 1 ⊢ 𝐵 = (𝐶 − 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5872 ℂcc 7806 + caddc 7811 − cmin 8124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-setind 4535 ax-resscn 7900 ax-1cn 7901 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-sub 8126 |
This theorem is referenced by: (None) |
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