Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
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 39229 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
2 | resubval 39272 | . . . . 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 2822 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) = 𝐵 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
5 | 4 | adantr 483 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ 𝐴) = 𝐵 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
6 | renegeu 39275 | . . . 4 ⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | |
7 | oveq2 7161 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵)) | |
8 | 7 | eqeq1d 2822 | . . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0)) |
9 | 8 | riota2 7136 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
10 | 6, 9 | sylan2 594 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
11 | 10 | ancoms 461 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) = 𝐵)) |
12 | 5, 11 | bitr4d 284 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃!wreu 3139 ℩crio 7110 (class class class)co 7153 ℝcr 10533 0cc0 10534 + caddc 10537 −ℝ cresub 39270 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-resscn 10591 ax-addrcl 10595 ax-rnegex 10605 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-po 5471 df-so 5472 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-ltxr 10677 df-resub 39271 |
This theorem is referenced by: renegid 39278 resubeulem1 39280 |
Copyright terms: Public domain | W3C validator |