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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubcan2 | Structured version Visualization version GIF version |
Description: Cancellation law for real subtraction. Compare subcan2 11481. (Contributed by Steven Nguyen, 8-Jan-2023.) |
Ref | Expression |
---|---|
resubcan2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) | |
2 | simpl1 1188 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → 𝐴 ∈ ℝ) | |
3 | simpl3 1190 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → 𝐶 ∈ ℝ) | |
4 | simpl2 1189 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → 𝐵 ∈ ℝ) | |
5 | rersubcl 41706 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 −ℝ 𝐶) ∈ ℝ) | |
6 | 4, 3, 5 | syl2anc 583 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → (𝐵 −ℝ 𝐶) ∈ ℝ) |
7 | 2, 3, 6 | resubaddd 41708 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) ↔ (𝐶 + (𝐵 −ℝ 𝐶)) = 𝐴)) |
8 | 1, 7 | mpbid 231 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → (𝐶 + (𝐵 −ℝ 𝐶)) = 𝐴) |
9 | repncan3 41711 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 + (𝐵 −ℝ 𝐶)) = 𝐵) | |
10 | 3, 4, 9 | syl2anc 583 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → (𝐶 + (𝐵 −ℝ 𝐶)) = 𝐵) |
11 | 8, 10 | eqtr3d 2766 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) → 𝐴 = 𝐵) |
12 | 11 | ex 412 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) → 𝐴 = 𝐵)) |
13 | oveq1 7408 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶)) | |
14 | 12, 13 | impbid1 224 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 (class class class)co 7401 ℝcr 11104 + caddc 11108 −ℝ cresub 41693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-addrcl 11166 ax-addass 11170 ax-rnegex 11176 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-resub 41694 |
This theorem is referenced by: (None) |
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