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| Mirrors > Home > MPE Home > Th. List > Mathboxes > resubsub4 | Structured version Visualization version GIF version | ||
| Description: Law for double subtraction. Compare subsub4 11543. (Contributed by Steven Nguyen, 14-Jan-2023.) | 
| Ref | Expression | 
|---|---|
| resubsub4 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) = (𝐴 −ℝ (𝐵 + 𝐶))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | readdcl 11239 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐶) ∈ ℝ) | |
| 2 | 1 | 3adant1 1130 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 + 𝐶) ∈ ℝ) | 
| 3 | rersubcl 42413 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) ∈ ℝ) | |
| 4 | 3 | 3adant3 1132 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 −ℝ 𝐵) ∈ ℝ) | 
| 5 | simp3 1138 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
| 6 | rersubcl 42413 | . . 3 ⊢ (((𝐴 −ℝ 𝐵) ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) ∈ ℝ) | |
| 7 | 4, 5, 6 | syl2anc 584 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) ∈ ℝ) | 
| 8 | simp2 1137 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℝ) | |
| 9 | 8 | recnd 11290 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ) | 
| 10 | 5 | recnd 11290 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ) | 
| 11 | 7 | recnd 11290 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) ∈ ℂ) | 
| 12 | 9, 10, 11 | addassd 11284 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐶) + ((𝐴 −ℝ 𝐵) −ℝ 𝐶)) = (𝐵 + (𝐶 + ((𝐴 −ℝ 𝐵) −ℝ 𝐶)))) | 
| 13 | repncan3 42418 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ (𝐴 −ℝ 𝐵) ∈ ℝ) → (𝐶 + ((𝐴 −ℝ 𝐵) −ℝ 𝐶)) = (𝐴 −ℝ 𝐵)) | |
| 14 | 5, 4, 13 | syl2anc 584 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 + ((𝐴 −ℝ 𝐵) −ℝ 𝐶)) = (𝐴 −ℝ 𝐵)) | 
| 15 | 14 | oveq2d 7448 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 + (𝐶 + ((𝐴 −ℝ 𝐵) −ℝ 𝐶))) = (𝐵 + (𝐴 −ℝ 𝐵))) | 
| 16 | simp1 1136 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℝ) | |
| 17 | repncan3 42418 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 + (𝐴 −ℝ 𝐵)) = 𝐴) | |
| 18 | 8, 16, 17 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 + (𝐴 −ℝ 𝐵)) = 𝐴) | 
| 19 | 12, 15, 18 | 3eqtrd 2780 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐵 + 𝐶) + ((𝐴 −ℝ 𝐵) −ℝ 𝐶)) = 𝐴) | 
| 20 | 2, 7, 19 | reladdrsub 42420 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) = (𝐴 −ℝ (𝐵 + 𝐶))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℝcr 11155 + caddc 11159 −ℝ cresub 42400 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-addrcl 11217 ax-addass 11221 ax-rnegex 11227 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-resub 42401 | 
| This theorem is referenced by: rennncan2 42425 repnpcan 42427 renegmulnnass 42488 | 
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