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Mirrors > Home > ILE Home > Th. List > divmulap | Unicode version |
Description: Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
Ref | Expression |
---|---|
divmulap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divvalap 8057 |
. . . . 5
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2 | 1 | 3expb 1142 |
. . . 4
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3 | 2 | 3adant2 960 |
. . 3
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4 | 3 | eqeq1d 2093 |
. 2
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5 | simp2 942 |
. . 3
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6 | receuap 8054 |
. . . . 5
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7 | 6 | 3expb 1142 |
. . . 4
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8 | 7 | 3adant2 960 |
. . 3
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9 | oveq2 5602 |
. . . . 5
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10 | 9 | eqeq1d 2093 |
. . . 4
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11 | 10 | riota2 5572 |
. . 3
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12 | 5, 8, 11 | syl2anc 403 |
. 2
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13 | 4, 12 | bitr4d 189 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-mulrcl 7365 ax-addcom 7366 ax-mulcom 7367 ax-addass 7368 ax-mulass 7369 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-1rid 7373 ax-0id 7374 ax-rnegex 7375 ax-precex 7376 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-apti 7381 ax-pre-ltadd 7382 ax-pre-mulgt0 7383 ax-pre-mulext 7384 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-id 4087 df-po 4090 df-iso 4091 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-iota 4937 df-fun 4974 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-reap 7970 df-ap 7977 df-div 8056 |
This theorem is referenced by: divmulap2 8059 divcanap2 8063 divrecap 8071 divcanap3 8081 div0ap 8085 div1 8086 recrecap 8092 rec11ap 8093 divdivdivap 8096 ddcanap 8109 rerecclap 8113 div2negap 8118 divmulapzi 8146 divmulapd 8194 caucvgrelemrec 10253 odd2np1 10667 sqgcd 10812 |
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