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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeu | Structured version Visualization version GIF version |
Description: Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.) |
Ref | Expression |
---|---|
resubeu | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
2 | rernegcl 39278 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ 𝐴) ∈ ℝ) |
4 | elre0re 39231 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ) | |
5 | 4, 4 | readdcld 10663 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 + 0) ∈ ℝ) |
6 | rernegcl 39278 | . . . . . . 7 ⊢ ((0 + 0) ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 + 0)) ∈ ℝ) |
8 | 7 | adantr 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 −ℝ (0 + 0)) ∈ ℝ) |
9 | simpr 487 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ) | |
10 | 8, 9 | readdcld 10663 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ (0 + 0)) + 𝐵) ∈ ℝ) |
11 | 3, 10 | readdcld 10663 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵)) ∈ ℝ) |
12 | resubeulem2 39283 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵))) = 𝐵) | |
13 | oveq2 7157 | . . . . 5 ⊢ (𝑥 = ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵)) → (𝐴 + 𝑥) = (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵)))) | |
14 | 13 | eqeq1d 2822 | . . . 4 ⊢ (𝑥 = ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵)) → ((𝐴 + 𝑥) = 𝐵 ↔ (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵))) = 𝐵)) |
15 | 14 | rspcev 3620 | . . 3 ⊢ ((((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵)) ∈ ℝ ∧ (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵))) = 𝐵) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵) |
16 | 11, 12, 15 | syl2anc 586 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵) |
17 | 1, 16 | renegeulem 39276 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ∃!wreu 3139 (class class class)co 7149 ℝcr 10529 0cc0 10530 + caddc 10533 −ℝ cresub 39272 |
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 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-addrcl 10591 ax-addass 10595 ax-rnegex 10601 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 |
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 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-ltxr 10673 df-resub 39273 |
This theorem is referenced by: rersubcl 39285 resubadd 39286 |
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