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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 10952. (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 10881 | . . 3 ⊢ (𝑥 = 𝐴 → -𝑥 = -𝐴) | |
2 | 1 | eleq1d 2900 | . 2 ⊢ (𝑥 = 𝐴 → (-𝑥 ∈ ℝ ↔ -𝐴 ∈ ℝ)) |
3 | vex 3500 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
4 | c0ex 10638 | . . . . . . 7 ⊢ 0 ∈ V | |
5 | 3, 4 | ifex 4518 | . . . . . 6 ⊢ if(𝑥 ∈ ℝ, 𝑥, 0) ∈ V |
6 | csbnegg 10886 | . . . . . 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 4386 | . . . . . . . . . . 11 ⊢ (0 ∈ V → ⦋0 / 𝑥⦌𝑥 = 0) | |
9 | 4, 8 | ax-mp 5 | . . . . . . . . . 10 ⊢ ⦋0 / 𝑥⦌𝑥 = 0 |
10 | 0re 10646 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
11 | 9, 10 | eqeltri 2912 | . . . . . . . . 9 ⊢ ⦋0 / 𝑥⦌𝑥 ∈ ℝ |
12 | sbcel1g 4368 | . . . . . . . . . 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 233 | . . . . . . . 8 ⊢ [0 / 𝑥]𝑥 ∈ ℝ |
15 | 14 | elimhyps 36101 | . . . . . . 7 ⊢ [if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]𝑥 ∈ ℝ |
16 | sbcel1g 4368 | . . . . . . . 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 232 | . . . . . 6 ⊢ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 ∈ ℝ |
19 | 18 | renegcli 10950 | . . . . 5 ⊢ -⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌𝑥 ∈ ℝ |
20 | 7, 19 | eqeltri 2912 | . . . 4 ⊢ ⦋if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥⦌-𝑥 ∈ ℝ |
21 | sbcel1g 4368 | . . . . 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 233 | . . 3 ⊢ [if(𝑥 ∈ ℝ, 𝑥, 0) / 𝑥]-𝑥 ∈ ℝ |
24 | 23 | dedths 36102 | . 2 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ) |
25 | 2, 24 | vtoclga 3577 | 1 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 Vcvv 3497 [wsbc 3775 ⦋csb 3886 ifcif 4470 ℝcr 10539 0cc0 10540 -cneg 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 |
This theorem is referenced by: (None) |
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