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Mirrors > Home > MPE Home > Th. List > rngorn1 | Unicode version |
Description: In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rnplrnml0.1 |
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rnplrnml0.2 |
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Ref | Expression |
---|---|
rngorn1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnplrnml0.2 |
. . . 4
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2 | 1 | rngogrpo 21935 |
. . 3
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3 | grporndm 21755 |
. . 3
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4 | 2, 3 | syl 16 |
. 2
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5 | rnplrnml0.1 |
. . 3
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6 | 5, 1 | rngodm1dm2 21963 |
. 2
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7 | 4, 6 | eqtrd 2440 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: rngomndo 21966 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-ral 2675 df-rex 2676 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-nul 3593 df-if 3704 df-sn 3784 df-pr 3785 df-op 3787 df-uni 3980 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-id 4462 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-fo 5423 df-fv 5425 df-ov 6047 df-1st 6312 df-2nd 6313 df-grpo 21736 df-ablo 21827 df-rngo 21921 |
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