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Mirrors > Home > MPE Home > Th. List > csbnegg | Structured version Visualization version GIF version |
Description: Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
csbnegg | ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbov2g 6691 | . 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(0 − 𝐵) = (0 − ⦋𝐴 / 𝑥⦌𝐵)) | |
2 | df-neg 10269 | . . 3 ⊢ -𝐵 = (0 − 𝐵) | |
3 | 2 | csbeq2i 3993 | . 2 ⊢ ⦋𝐴 / 𝑥⦌-𝐵 = ⦋𝐴 / 𝑥⦌(0 − 𝐵) |
4 | df-neg 10269 | . 2 ⊢ -⦋𝐴 / 𝑥⦌𝐵 = (0 − ⦋𝐴 / 𝑥⦌𝐵) | |
5 | 1, 3, 4 | 3eqtr4g 2681 | 1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌-𝐵 = -⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⦋csb 3533 (class class class)co 6650 0cc0 9936 − cmin 10266 -cneg 10267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 df-neg 10269 |
This theorem is referenced by: dvfsum2 23797 renegclALT 34249 |
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