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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 11502. (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 1134 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
2 | rersubcl 41553 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐶 −ℝ 𝐴) ∈ ℝ) | |
3 | 2 | ancoms 457 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 −ℝ 𝐴) ∈ ℝ) |
4 | 3 | 3adant2 1129 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 −ℝ 𝐴) ∈ ℝ) |
5 | simp2 1135 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) | |
6 | readdsub 41559 | . . 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 1369 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + (𝐶 −ℝ 𝐴)) −ℝ 𝐵) = ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴))) |
8 | repncan3 41558 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + (𝐶 −ℝ 𝐴)) = 𝐶) | |
9 | 8 | 3adant2 1129 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 + (𝐶 −ℝ 𝐴)) = 𝐶) |
10 | 9 | oveq1d 7426 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + (𝐶 −ℝ 𝐴)) −ℝ 𝐵) = (𝐶 −ℝ 𝐵)) |
11 | 7, 10 | eqtr3d 2772 | 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 1085 = wceq 1539 ∈ wcel 2104 (class class class)co 7411 ℝcr 11111 + caddc 11115 −ℝ cresub 41540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-resscn 11169 ax-addrcl 11173 ax-addass 11177 ax-rnegex 11183 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-resub 41541 |
This theorem is referenced by: (None) |
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