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Mirrors > Home > ILE Home > Th. List > subaddeqd | GIF version |
Description: Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
Ref | Expression |
---|---|
subaddeqd.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
subaddeqd.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
subaddeqd.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
subaddeqd.d | ⊢ (𝜑 → 𝐷 ∈ ℂ) |
subaddeqd.1 | ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) |
Ref | Expression |
---|---|
subaddeqd | ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subaddeqd.1 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷)) | |
2 | 1 | oveq1d 5789 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = ((𝐶 + 𝐷) − (𝐷 + 𝐵))) |
3 | subaddeqd.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | subaddeqd.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℂ) | |
5 | 3, 4 | addcomd 7913 | . . . 4 ⊢ (𝜑 → (𝐶 + 𝐷) = (𝐷 + 𝐶)) |
6 | 5 | oveq1d 5789 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐷) − (𝐷 + 𝐵)) = ((𝐷 + 𝐶) − (𝐷 + 𝐵))) |
7 | 2, 6 | eqtrd 2172 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = ((𝐷 + 𝐶) − (𝐷 + 𝐵))) |
8 | subaddeqd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
9 | subaddeqd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
10 | 8, 4, 9 | pnpcan2d 8111 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) − (𝐷 + 𝐵)) = (𝐴 − 𝐷)) |
11 | 4, 3, 9 | pnpcand 8110 | . 2 ⊢ (𝜑 → ((𝐷 + 𝐶) − (𝐷 + 𝐵)) = (𝐶 − 𝐵)) |
12 | 7, 10, 11 | 3eqtr3d 2180 | 1 ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐶 − 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 + caddc 7623 − cmin 7933 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 |
This theorem is referenced by: (None) |
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