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Mirrors > Home > MPE Home > Th. List > addlsub | Structured version Visualization version 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 6929 | . . 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 10735 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
5 | eqtr2 2799 | . . . . . 6 ⊢ ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → (𝐶 − 𝐵) = 𝐴) | |
6 | 5 | eqcomd 2783 | . . . . 5 ⊢ ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → 𝐴 = (𝐶 − 𝐵)) |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → ((((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) ∧ ((𝐴 + 𝐵) − 𝐵) = 𝐴) → 𝐴 = (𝐶 − 𝐵))) |
8 | 4, 7 | mpan2d 684 | . . 3 ⊢ (𝜑 → (((𝐴 + 𝐵) − 𝐵) = (𝐶 − 𝐵) → 𝐴 = (𝐶 − 𝐵))) |
9 | 1, 8 | syl5 34 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 → 𝐴 = (𝐶 − 𝐵))) |
10 | oveq1 6929 | . . 3 ⊢ (𝐴 = (𝐶 − 𝐵) → (𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵)) | |
11 | addlsub.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
12 | 11, 3 | npcand 10738 | . . . 4 ⊢ (𝜑 → ((𝐶 − 𝐵) + 𝐵) = 𝐶) |
13 | eqtr 2798 | . . . . 5 ⊢ (((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) ∧ ((𝐶 − 𝐵) + 𝐵) = 𝐶) → (𝐴 + 𝐵) = 𝐶) | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) ∧ ((𝐶 − 𝐵) + 𝐵) = 𝐶) → (𝐴 + 𝐵) = 𝐶)) |
15 | 12, 14 | mpan2d 684 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐵) = ((𝐶 − 𝐵) + 𝐵) → (𝐴 + 𝐵) = 𝐶)) |
16 | 10, 15 | syl5 34 | . 2 ⊢ (𝜑 → (𝐴 = (𝐶 − 𝐵) → (𝐴 + 𝐵) = 𝐶)) |
17 | 9, 16 | impbid 204 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) = 𝐶 ↔ 𝐴 = (𝐶 − 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 (class class class)co 6922 ℂcc 10270 + caddc 10275 − cmin 10606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 |
This theorem is referenced by: addrsub 10792 subexsub 10793 lineq 11212 nn0ob 15514 blen1b 43390 nn0sumshdiglem1 43423 |
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