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Mirrors > Home > MPE Home > Th. List > Mathboxes > renegclALT | Structured version Visualization version GIF version |
Description: Closure law for negative of reals. Demonstrates use of weak deduction theorem with explicit substitution. The proof is much longer than that of renegcl 11595. (Contributed by NM, 15-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
renegclALT | ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negeq 11524 | . . 3 ⊢ (𝑥 = 𝐴 → -𝑥 = -𝐴) | |
2 | 1 | eleq1d 2823 | . 2 ⊢ (𝑥 = 𝐴 → (-𝑥 ∈ ℝ ↔ -𝐴 ∈ ℝ)) |
3 | vex 3486 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | c0ex 11280 | . . . . . . 7 ⊢ 0 ∈ V | |
5 | 3, 4 | ifex 4598 | . . . . . 6 ⊢ if(𝑥 ∈ ℝ, 𝑥, 0) ∈ V |
6 | csbnegg 11529 | . . . . . 6 ⊢ (if(𝑥 ∈ ℝ, 𝑥, 0) ∈ V → ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌-𝑥 = -⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥) | |
7 | 5, 6 | ax-mp 5 | . . . . 5 ⊢ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌-𝑥 = -⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 |
8 | csbvarg 4453 | . . . . . . . . . . 11 ⊢ (0 ∈ V → ⦋0 / 𝑥⦌𝑥 = 0) | |
9 | 4, 8 | ax-mp 5 | . . . . . . . . . 10 ⊢ ⦋0 / 𝑥⦌𝑥 = 0 |
10 | 0re 11288 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
11 | 9, 10 | eqeltri 2834 | . . . . . . . . 9 ⊢ ⦋0 / 𝑥⦌𝑥 ∈ ℝ |
12 | sbcel1g 4435 | . . . . . . . . . 10 ⊢ (0 ∈ V → ([0 / 𝑥]𝑥 ∈ ℝ ↔ ⦋0 / 𝑥⦌𝑥 ∈ ℝ)) | |
13 | 4, 12 | ax-mp 5 | . . . . . . . . 9 ⊢ ([0 / 𝑥]𝑥 ∈ ℝ ↔ ⦋0 / 𝑥⦌𝑥 ∈ ℝ) |
14 | 11, 13 | mpbir 231 | . . . . . . . 8 ⊢ [0 / 𝑥]𝑥 ∈ ℝ |
15 | 14 | elimhyps 38866 | . . . . . . 7 ⊢ [if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]𝑥 ∈ ℝ |
16 | sbcel1g 4435 | . . . . . . . 8 ⊢ (if(𝑥 ∈ ℝ, 𝑥, 0) ∈ V → ([if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]𝑥 ∈ ℝ ↔ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 ∈ ℝ)) | |
17 | 5, 16 | ax-mp 5 | . . . . . . 7 ⊢ ([if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]𝑥 ∈ ℝ ↔ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 ∈ ℝ) |
18 | 15, 17 | mpbi 230 | . . . . . 6 ⊢ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 ∈ ℝ |
19 | 18 | renegcli 11593 | . . . . 5 ⊢ -⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 ∈ ℝ |
20 | 7, 19 | eqeltri 2834 | . . . 4 ⊢ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌-𝑥 ∈ ℝ |
21 | sbcel1g 4435 | . . . . 5 ⊢ (if(𝑥 ∈ ℝ, 𝑥, 0) ∈ V → ([if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]-𝑥 ∈ ℝ ↔ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌-𝑥 ∈ ℝ)) | |
22 | 5, 21 | ax-mp 5 | . . . 4 ⊢ ([if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]-𝑥 ∈ ℝ ↔ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌-𝑥 ∈ ℝ) |
23 | 20, 22 | mpbir 231 | . . 3 ⊢ [if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]-𝑥 ∈ ℝ |
24 | 23 | dedths 38867 | . 2 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) |
25 | 2, 24 | vtoclga 3585 | 1 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2103 Vcvv 3482 [wsbc 3798 ⦋csb 3915 ifcif 4548 ℝcr 11179 0cc0 11180 -cneg 11517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-ltxr 11325 df-sub 11518 df-neg 11519 |
This theorem is referenced by: (None) |
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