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Mirrors > Home > MPE Home > Th. List > Mathboxes > subresre | Structured version Visualization version GIF version |
Description: Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.) |
Ref | Expression |
---|---|
subresre | ⊢ −ℝ = ( − ↾ (ℝ × ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubeqsub 39893 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 −ℝ 𝑦) = (𝑥 − 𝑦)) | |
2 | 1 | 3adant1 1128 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 −ℝ 𝑦) = (𝑥 − 𝑦)) |
3 | ax-resscn 10617 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ℝ ⊆ ℂ) |
5 | resubf 39846 | . . . 4 ⊢ −ℝ :(ℝ × ℝ)⟶ℝ | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → −ℝ :(ℝ × ℝ)⟶ℝ) |
7 | sn-subf 39892 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → − :(ℂ × ℂ)⟶ℂ) |
9 | 2, 4, 6, 8 | oprres 7305 | . 2 ⊢ (⊤ → −ℝ = ( − ↾ (ℝ × ℝ))) |
10 | 9 | mptru 1546 | 1 ⊢ −ℝ = ( − ↾ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊤wtru 1540 ∈ wcel 2112 ⊆ wss 3854 × cxp 5515 ↾ cres 5519 ⟶wf 6324 (class class class)co 7143 ℂcc 10558 ℝcr 10559 − cmin 10893 −ℝ cresub 39830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-po 5436 df-so 5437 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-1st 7686 df-2nd 7687 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-ltxr 10703 df-sub 10895 df-2 11722 df-3 11723 df-resub 39831 |
This theorem is referenced by: (None) |
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