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Mirrors > Home > ILE Home > Th. List > subaddrii | GIF version |
Description: Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
pncan3i.2 | ⊢ 𝐵 ∈ ℂ |
subadd.3 | ⊢ 𝐶 ∈ ℂ |
subaddri.4 | ⊢ (𝐵 + 𝐶) = 𝐴 |
Ref | Expression |
---|---|
subaddrii | ⊢ (𝐴 − 𝐵) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subaddri.4 | . 2 ⊢ (𝐵 + 𝐶) = 𝐴 | |
2 | negidi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | pncan3i.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | subadd.3 | . . 3 ⊢ 𝐶 ∈ ℂ | |
5 | 2, 3, 4 | subaddi 7767 | . 2 ⊢ ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴) |
6 | 1, 5 | mpbir 144 | 1 ⊢ (𝐴 − 𝐵) = 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ∈ wcel 1438 (class class class)co 5652 ℂcc 7346 + caddc 7351 − cmin 7651 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-setind 4353 ax-resscn 7435 ax-1cn 7436 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-addcom 7443 ax-addass 7445 ax-distr 7447 ax-i2m1 7448 ax-0id 7451 ax-rnegex 7452 ax-cnre 7454 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-sub 7653 |
This theorem is referenced by: 2m1e1 8538 3m1e2 8540 fzo0to42pr 9627 4bc3eq4 10177 4bc2eq6 10178 cos1bnd 11046 cos2bnd 11047 |
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