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 2823 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
2 | iccssxr 12822 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
3 | xrsbas 20563 | . . . 4 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | sseqtri 4005 | . . 3 ⊢ (0[,]+∞) ⊆ (Base‘ℝ*𝑠) |
5 | eqid 2823 | . . 3 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠) | |
6 | eqid 2823 | . . 3 ⊢ (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠) | |
7 | xrs0 30664 | . . . 4 ⊢ 0 = (0g‘ℝ*𝑠) | |
8 | xrge00 30675 | . . . 4 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
9 | 7, 8 | eqtr3i 2848 | . . 3 ⊢ (0g‘ℝ*𝑠) = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 1, 4, 5, 6, 9 | ressmulgnn0 30673 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴(.g‘ℝ*𝑠)𝐵)) |
11 | nn0z 12008 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
12 | eliccxr 12826 | . . 3 ⊢ (𝐵 ∈ (0[,]+∞) → 𝐵 ∈ ℝ*) | |
13 | xrsmulgzz 30667 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)) | |
14 | 11, 12, 13 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)) |
15 | 10, 14 | eqtrd 2858 | 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 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 ℝ*cxr 10676 ℕ0cn0 11900 ℤcz 11984 ·e cxmu 12509 [,]cicc 12744 Basecbs 16485 ↾s cress 16486 0gc0g 16715 ℝ*𝑠cxrs 16775 invgcminusg 18106 .gcmg 18226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-seq 13373 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-tset 16586 df-ple 16587 df-ds 16589 df-0g 16717 df-xrs 16777 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-minusg 18109 df-mulg 18227 df-cmn 18910 |
This theorem is referenced by: esumcst 31324 |
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