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Mirrors > Home > ILE Home > Th. List > addlsub | GIF version |
Description: Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.) |
Ref | Expression |
---|---|
addlsub.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addlsub.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addlsub.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
addlsub | ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 − 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5879 | . . 3 ⊢ ((𝐴 + 𝐵) = 𝐶 → ((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵)) | |
2 | addlsub.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | addlsub.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | 2, 3 | pncand 8265 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
5 | eqtr2 2196 | . . . . . 6 ⊢ ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → (𝐶 − 𝐵) = 𝐴) | |
6 | 5 | eqcomd 2183 | . . . . 5 ⊢ ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → 𝐴 = (𝐶 − 𝐵)) |
7 | 6 | a1i 9 | . . . 4 ⊢ (𝜑 → ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → 𝐴 = (𝐶 − 𝐵))) |
8 | 4, 7 | mpan2d 428 | . . 3 ⊢ (𝜑 → (((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) → 𝐴 = (𝐶 − 𝐵))) |
9 | 1, 8 | syl5 32 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 → 𝐴 = (𝐶 − 𝐵))) |
10 | oveq1 5879 | . . 3 ⊢ (𝐴 = (𝐶 − 𝐵) → (𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵)) | |
11 | addlsub.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
12 | 11, 3 | npcand 8268 | . . . 4 ⊢ (𝜑 → ((𝐶 − 𝐵) + 𝐵) = 𝐶) |
13 | eqtr 2195 | . . . . 5 ⊢ (((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) ∧ ((𝐶 − 𝐵) + 𝐵) = 𝐶) → (𝐴 + 𝐵) = 𝐶) | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝜑 → (((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) ∧ ((𝐶 − 𝐵) + 𝐵) = 𝐶) → (𝐴 + 𝐵) = 𝐶)) |
15 | 12, 14 | mpan2d 428 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) → (𝐴 + 𝐵) = 𝐶)) |
16 | 10, 15 | syl5 32 | . 2 ⊢ (𝜑 → (𝐴 = (𝐶 − 𝐵) → (𝐴 + 𝐵) = 𝐶)) |
17 | 9, 16 | impbid 129 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 − 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = 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: addrsub 8324 subexsub 8325 nn0ob 11905 oddennn 12385 |
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