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Mirrors > Home > MPE Home > Th. List > Mathboxes > renegadd | Structured version Visualization version GIF version |
Description: Relationship between real negation and addition. (Contributed by Steven Nguyen, 7-Jan-2023.) |
Ref | Expression |
---|---|
renegadd | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elre0re 41998 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | resubval 42076 | . . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 −ℝ 𝐴) = (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)) | |
3 | 1, 2 | mpancom 686 | . . . 4 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) = (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)) |
4 | 3 | eqeq1d 2728 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) = 𝐵 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
5 | 4 | adantr 479 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ 𝐴) = 𝐵 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
6 | renegeu 42079 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | |
7 | oveq2 7422 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵)) | |
8 | 7 | eqeq1d 2728 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0)) |
9 | 8 | riota2 7396 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
10 | 6, 9 | sylan2 591 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
11 | 10 | ancoms 457 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
12 | 5, 11 | bitr4d 281 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃!wreu 3363 ℩crio 7369 (class class class)co 7414 ℝcr 11146 0cc0 11147 + caddc 11150 −ℝ cresub 42074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-resscn 11204 ax-addrcl 11208 ax-rnegex 11218 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-ltxr 11292 df-resub 42075 |
This theorem is referenced by: renegid 42082 resubeulem1 42084 sn-inelr 42176 |
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