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Mirrors > Home > MPE Home > Th. List > xrsmcmn | Structured version Unicode version |
Description: The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrs1mnd 17977.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
xrsmcmn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2454 |
. . . . 5
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2 | xrsbas 17958 |
. . . . 5
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3 | 1, 2 | mgpbas 16720 |
. . . 4
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4 | 3 | a1i 11 |
. . 3
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5 | xrsmul 17960 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 1, 5 | mgpplusg 16718 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 6 | a1i 11 |
. . 3
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8 | xmulcl 11348 |
. . . . 5
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9 | 8 | 3adant1 1006 |
. . . 4
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10 | xmulass 11362 |
. . . . 5
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11 | 10 | adantl 466 |
. . . 4
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12 | 1re 9497 |
. . . . 5
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13 | rexr 9541 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 12, 13 | mp1i 12 |
. . . 4
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15 | xmulid2 11355 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | adantl 466 |
. . . 4
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17 | xmulid1 11354 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
18 | 17 | adantl 466 |
. . . 4
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19 | 4, 7, 9, 11, 14, 16, 18 | ismndd 15564 |
. . 3
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20 | xmulcom 11341 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 20 | 3adant1 1006 |
. . 3
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22 | 4, 7, 19, 21 | iscmnd 16411 |
. 2
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23 | 22 | trud 1379 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-cnex 9450 ax-resscn 9451 ax-1cn 9452 ax-icn 9453 ax-addcl 9454 ax-addrcl 9455 ax-mulcl 9456 ax-mulrcl 9457 ax-mulcom 9458 ax-addass 9459 ax-mulass 9460 ax-distr 9461 ax-i2m1 9462 ax-1ne0 9463 ax-1rid 9464 ax-rnegex 9465 ax-rrecex 9466 ax-cnre 9467 ax-pre-lttri 9468 ax-pre-lttrn 9469 ax-pre-ltadd 9470 ax-pre-mulgt0 9471 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-pss 3453 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-tp 3991 df-op 3993 df-uni 4201 df-int 4238 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-tr 4495 df-eprel 4741 df-id 4745 df-po 4750 df-so 4751 df-fr 4788 df-we 4790 df-ord 4831 df-on 4832 df-lim 4833 df-suc 4834 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-riota 6162 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-om 6588 df-1st 6688 df-2nd 6689 df-recs 6943 df-rdg 6977 df-1o 7031 df-oadd 7035 df-er 7212 df-en 7422 df-dom 7423 df-sdom 7424 df-fin 7425 df-pnf 9532 df-mnf 9533 df-xr 9534 df-ltxr 9535 df-le 9536 df-sub 9709 df-neg 9710 df-nn 10435 df-2 10492 df-3 10493 df-4 10494 df-5 10495 df-6 10496 df-7 10497 df-8 10498 df-9 10499 df-10 10500 df-n0 10692 df-z 10759 df-dec 10868 df-uz 10974 df-xneg 11201 df-xmul 11203 df-fz 11556 df-struct 14295 df-ndx 14296 df-slot 14297 df-base 14298 df-sets 14299 df-plusg 14371 df-mulr 14372 df-tset 14377 df-ple 14378 df-ds 14380 df-xrs 14560 df-mnd 15535 df-cmn 16401 df-mgp 16715 |
This theorem is referenced by: (None) |
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