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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 11601. (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 11530 | . . 3 ⊢ (𝑥 = 𝐴 → -𝑥 = -𝐴) | |
2 | 1 | eleq1d 2829 | . 2 ⊢ (𝑥 = 𝐴 → (-𝑥 ∈ ℝ ↔ -𝐴 ∈ ℝ)) |
3 | vex 3492 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | c0ex 11286 | . . . . . . 7 ⊢ 0 ∈ V | |
5 | 3, 4 | ifex 4598 | . . . . . 6 ⊢ if(𝑥 ∈ ℝ, 𝑥, 0) ∈ V |
6 | csbnegg 11535 | . . . . . 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 4457 | . . . . . . . . . . 11 ⊢ (0 ∈ V → ⦋0 / 𝑥⦌𝑥 = 0) | |
9 | 4, 8 | ax-mp 5 | . . . . . . . . . 10 ⊢ ⦋0 / 𝑥⦌𝑥 = 0 |
10 | 0re 11294 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
11 | 9, 10 | eqeltri 2840 | . . . . . . . . 9 ⊢ ⦋0 / 𝑥⦌𝑥 ∈ ℝ |
12 | sbcel1g 4439 | . . . . . . . . . 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 38919 | . . . . . . 7 ⊢ [if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]𝑥 ∈ ℝ |
16 | sbcel1g 4439 | . . . . . . . 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 11599 | . . . . 5 ⊢ -⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 ∈ ℝ |
20 | 7, 19 | eqeltri 2840 | . . . 4 ⊢ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌-𝑥 ∈ ℝ |
21 | sbcel1g 4439 | . . . . 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 38920 | . 2 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) |
25 | 2, 24 | vtoclga 3589 | 1 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 Vcvv 3488 [wsbc 3804 ⦋csb 3921 ifcif 4548 ℝcr 11185 0cc0 11186 -cneg 11523 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-ltxr 11331 df-sub 11524 df-neg 11525 |
This theorem is referenced by: (None) |
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