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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 40884 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 −ℝ 𝑦) = (𝑥 − 𝑦)) | |
2 | 1 | 3adant1 1130 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 −ℝ 𝑦) = (𝑥 − 𝑦)) |
3 | ax-resscn 11108 | . . . 4 ⊢ ℝ ⊆ ℂ | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → ℝ ⊆ ℂ) |
5 | resubf 40836 | . . . 4 ⊢ −ℝ :(ℝ × ℝ)⟶ℝ | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → −ℝ :(ℝ × ℝ)⟶ℝ) |
7 | sn-subf 40883 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → − :(ℂ × ℂ)⟶ℂ) |
9 | 2, 4, 6, 8 | oprres 7522 | . 2 ⊢ (⊤ → −ℝ = ( − ↾ (ℝ × ℝ))) |
10 | 9 | mptru 1548 | 1 ⊢ −ℝ = ( − ↾ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ⊆ wss 3910 × cxp 5631 ↾ cres 5635 ⟶wf 6492 (class class class)co 7357 ℂcc 11049 ℝcr 11050 − cmin 11385 −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-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 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-sub 11387 df-2 12216 df-3 12217 df-resub 40821 |
This theorem is referenced by: (None) |
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