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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sgnmulrp2 | Structured version Visualization version GIF version |
Description: Multiplication by a positive number does not affect signum. (Contributed by Thierry Arnoux, 2-Oct-2018.) |
Ref | Expression |
---|---|
sgnmulrp2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (sgn‘(𝐴 · 𝐵)) = (sgn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ+) | |
2 | 1 | rpred 12241 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ) |
3 | sgnmul 31403 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵))) | |
4 | 2, 3 | syldan 582 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (sgn‘(𝐴 · 𝐵)) = ((sgn‘𝐴) · (sgn‘𝐵))) |
5 | 1 | rpxrd 12242 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℝ*) |
6 | 1 | rpgt0d 12244 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 0 < 𝐵) |
7 | sgnp 14300 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 0 < 𝐵) → (sgn‘𝐵) = 1) | |
8 | 5, 6, 7 | syl2anc 576 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (sgn‘𝐵) = 1) |
9 | 8 | oveq2d 6986 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((sgn‘𝐴) · (sgn‘𝐵)) = ((sgn‘𝐴) · 1)) |
10 | simpl 475 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 ∈ ℝ) | |
11 | sgnclre 31400 | . . 3 ⊢ (𝐴 ∈ ℝ → (sgn‘𝐴) ∈ ℝ) | |
12 | ax-1rid 10397 | . . 3 ⊢ ((sgn‘𝐴) ∈ ℝ → ((sgn‘𝐴) · 1) = (sgn‘𝐴)) | |
13 | 10, 11, 12 | 3syl 18 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → ((sgn‘𝐴) · 1) = (sgn‘𝐴)) |
14 | 4, 9, 13 | 3eqtrd 2812 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (sgn‘(𝐴 · 𝐵)) = (sgn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 class class class wbr 4923 ‘cfv 6182 (class class class)co 6970 ℝcr 10326 0cc0 10327 1c1 10328 · cmul 10332 ℝ*cxr 10465 < clt 10466 ℝ+crp 12197 sgncsgn 14296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-rp 12198 df-sgn 14297 |
This theorem is referenced by: (None) |
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