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Mirrors > Home > ILE Home > Th. List > mulclpr | Unicode version |
Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) |
Ref | Expression |
---|---|
mulclpr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-imp 7089 |
. . . 4
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2 | 1 | genpelxp 7131 |
. . 3
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3 | mulclnq 6996 |
. . . 4
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4 | 1, 3 | genpml 7137 |
. . 3
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5 | 1, 3 | genpmu 7138 |
. . 3
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6 | 2, 4, 5 | jca32 304 |
. 2
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7 | ltmnqg 7021 |
. . . . 5
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8 | mulcomnqg 7003 |
. . . . 5
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9 | mulnqprl 7188 |
. . . . 5
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10 | 1, 3, 7, 8, 9 | genprndl 7141 |
. . . 4
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11 | mulnqpru 7189 |
. . . . 5
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12 | 1, 3, 7, 8, 11 | genprndu 7142 |
. . . 4
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13 | 10, 12 | jca 301 |
. . 3
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14 | 1, 3, 7, 8 | genpdisj 7143 |
. . 3
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15 | mullocpr 7191 |
. . 3
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16 | 13, 14, 15 | 3jca 1124 |
. 2
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17 | elnp1st2nd 7096 |
. 2
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18 | 6, 16, 17 | sylanbrc 409 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-eprel 4125 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-2o 6196 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6924 df-pli 6925 df-mi 6926 df-lti 6927 df-plpq 6964 df-mpq 6965 df-enq 6967 df-nqqs 6968 df-plqqs 6969 df-mqqs 6970 df-1nqqs 6971 df-rq 6972 df-ltnqqs 6973 df-enq0 7044 df-nq0 7045 df-0nq0 7046 df-plq0 7047 df-mq0 7048 df-inp 7086 df-imp 7089 |
This theorem is referenced by: mulnqprlemfl 7195 mulnqprlemfu 7196 mulnqpr 7197 mulassprg 7201 distrlem1prl 7202 distrlem1pru 7203 distrlem4prl 7204 distrlem4pru 7205 distrlem5prl 7206 distrlem5pru 7207 distrprg 7208 1idpr 7212 recexprlemex 7257 ltmprr 7262 mulcmpblnrlemg 7347 mulcmpblnr 7348 mulclsr 7361 mulcomsrg 7364 mulasssrg 7365 distrsrg 7366 m1m1sr 7368 1idsr 7375 00sr 7376 recexgt0sr 7380 mulgt0sr 7384 mulextsr1lem 7386 mulextsr1 7387 recidpirq 7456 |
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