Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0mulgnn0 | Structured version Visualization version GIF version |
Description: The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
Ref | Expression |
---|---|
xrge0mulgnn0 | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
2 | iccssxr 13018 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
3 | xrsbas 20379 | . . . 4 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | sseqtri 3937 | . . 3 ⊢ (0[,]+∞) ⊆ (Base‘ℝ*𝑠) |
5 | eqid 2737 | . . 3 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠) | |
6 | eqid 2737 | . . 3 ⊢ (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠) | |
7 | xrs0 31003 | . . . 4 ⊢ 0 = (0g‘ℝ*𝑠) | |
8 | xrge00 31014 | . . . 4 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
9 | 7, 8 | eqtr3i 2767 | . . 3 ⊢ (0g‘ℝ*𝑠) = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 1, 4, 5, 6, 9 | ressmulgnn0 31012 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴(.g‘ℝ*𝑠)𝐵)) |
11 | nn0z 12200 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
12 | eliccxr 13023 | . . 3 ⊢ (𝐵 ∈ (0[,]+∞) → 𝐵 ∈ ℝ*) | |
13 | xrsmulgzz 31006 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)) | |
14 | 11, 12, 13 | syl2an 599 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)) |
15 | 10, 14 | eqtrd 2777 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 0cc0 10729 +∞cpnf 10864 ℝ*cxr 10866 ℕ0cn0 12090 ℤcz 12176 ·e cxmu 12703 [,]cicc 12938 Basecbs 16760 ↾s cress 16784 0gc0g 16944 ℝ*𝑠cxrs 17005 invgcminusg 18366 .gcmg 18488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-seq 13575 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-tset 16821 df-ple 16822 df-ds 16824 df-0g 16946 df-xrs 17007 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-minusg 18369 df-mulg 18489 df-cmn 19172 |
This theorem is referenced by: esumcst 31743 |
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