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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 2728 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
2 | iccssxr 13433 | . . . 4 ⊢ (0[,]+∞) ⊆ ℝ* | |
3 | xrsbas 21304 | . . . 4 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
4 | 2, 3 | sseqtri 4014 | . . 3 ⊢ (0[,]+∞) ⊆ (Base‘ℝ*𝑠) |
5 | eqid 2728 | . . 3 ⊢ (.g‘ℝ*𝑠) = (.g‘ℝ*𝑠) | |
6 | eqid 2728 | . . 3 ⊢ (invg‘ℝ*𝑠) = (invg‘ℝ*𝑠) | |
7 | xrs0 32727 | . . . 4 ⊢ 0 = (0g‘ℝ*𝑠) | |
8 | xrge00 32736 | . . . 4 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
9 | 7, 8 | eqtr3i 2758 | . . 3 ⊢ (0g‘ℝ*𝑠) = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 1, 4, 5, 6, 9 | ressmulgnn0 19026 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴(.g‘ℝ*𝑠)𝐵)) |
11 | nn0z 12607 | . . 3 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
12 | eliccxr 13438 | . . 3 ⊢ (𝐵 ∈ (0[,]+∞) → 𝐵 ∈ ℝ*) | |
13 | xrsmulgzz 32730 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)) | |
14 | 11, 12, 13 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵)) |
15 | 10, 14 | eqtrd 2768 | 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 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 0cc0 11132 +∞cpnf 11269 ℝ*cxr 11271 ℕ0cn0 12496 ℤcz 12582 ·e cxmu 13117 [,]cicc 13353 Basecbs 17173 ↾s cress 17202 0gc0g 17414 ℝ*𝑠cxrs 17475 invgcminusg 18884 .gcmg 19016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-icc 13357 df-fz 13511 df-seq 13993 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-tset 17245 df-ple 17246 df-ds 17248 df-0g 17416 df-xrs 17477 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-minusg 18887 df-mulg 19017 df-cmn 19730 |
This theorem is referenced by: esumcst 33676 |
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