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Mirrors > Home > MPE Home > Th. List > Mathboxes > renpncan3 | Structured version Visualization version GIF version |
Description: Cancellation law for real subtraction. Compare npncan3 11494. (Contributed by Steven Nguyen, 28-Jan-2023.) |
Ref | Expression |
---|---|
renpncan3 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴)) = (𝐶 −ℝ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1137 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
2 | rersubcl 41195 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 −ℝ 𝐴) ∈ ℝ) | |
3 | 2 | ancoms 460 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 −ℝ 𝐴) ∈ ℝ) |
4 | 3 | 3adant2 1132 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 −ℝ 𝐴) ∈ ℝ) |
5 | simp2 1138 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) | |
6 | readdsub 41201 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 −ℝ 𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + (𝐶 −ℝ 𝐴)) −ℝ 𝐵) = ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴))) | |
7 | 1, 4, 5, 6 | syl3anc 1372 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + (𝐶 −ℝ 𝐴)) −ℝ 𝐵) = ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴))) |
8 | repncan3 41200 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + (𝐶 −ℝ 𝐴)) = 𝐶) | |
9 | 8 | 3adant2 1132 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + (𝐶 −ℝ 𝐴)) = 𝐶) |
10 | 9 | oveq1d 7419 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + (𝐶 −ℝ 𝐴)) −ℝ 𝐵) = (𝐶 −ℝ 𝐵)) |
11 | 7, 10 | eqtr3d 2775 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴)) = (𝐶 −ℝ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 (class class class)co 7404 ℝcr 11105 + caddc 11109 −ℝ cresub 41182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-addrcl 11167 ax-addass 11171 ax-rnegex 11177 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-resub 41183 |
This theorem is referenced by: (None) |
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