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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubf | Structured version Visualization version GIF version |
Description: Real subtraction is an operation on the real numbers. Based on subf 11403. (Contributed by Steven Nguyen, 7-Jan-2023.) |
Ref | Expression |
---|---|
resubf | ⊢ −ℝ :(ℝ × ℝ)⟶ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubval 40822 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 −ℝ 𝑦) = (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)) | |
2 | rersubcl 40833 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 −ℝ 𝑦) ∈ ℝ) | |
3 | 1, 2 | eqeltrrd 2839 | . . 3 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥) ∈ ℝ) |
4 | 3 | rgen2 3194 | . 2 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥) ∈ ℝ |
5 | df-resub 40821 | . . 3 ⊢ −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)) | |
6 | 5 | fmpo 8000 | . 2 ⊢ (∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥) ∈ ℝ ↔ −ℝ :(ℝ × ℝ)⟶ℝ) |
7 | 4, 6 | mpbi 229 | 1 ⊢ −ℝ :(ℝ × ℝ)⟶ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 × cxp 5631 ⟶wf 6492 ℩crio 7312 (class class class)co 7357 ℝcr 11050 + caddc 11054 −ℝ cresub 40820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-addrcl 11112 ax-addass 11116 ax-rnegex 11122 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-ltxr 11194 df-resub 40821 |
This theorem is referenced by: subresre 40885 |
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