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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngorn1eq | Structured version Visualization version GIF version |
Description: In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rnplrnml0.1 | β’ π» = (2nd βπ ) |
rnplrnml0.2 | β’ πΊ = (1st βπ ) |
Ref | Expression |
---|---|
rngorn1eq | β’ (π β RingOps β ran πΊ = ran π») |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnplrnml0.2 | . . . 4 β’ πΊ = (1st βπ ) | |
2 | rnplrnml0.1 | . . . 4 β’ π» = (2nd βπ ) | |
3 | eqid 2728 | . . . 4 β’ ran πΊ = ran πΊ | |
4 | 1, 2, 3 | rngosm 37373 | . . 3 β’ (π β RingOps β π»:(ran πΊ Γ ran πΊ)βΆran πΊ) |
5 | 1, 2, 3 | rngoi 37372 | . . . 4 β’ (π β RingOps β ((πΊ β AbelOp β§ π»:(ran πΊ Γ ran πΊ)βΆran πΊ) β§ (βπ₯ β ran πΊβπ¦ β ran πΊβπ§ β ran πΊ(((π₯π»π¦)π»π§) = (π₯π»(π¦π»π§)) β§ (π₯π»(π¦πΊπ§)) = ((π₯π»π¦)πΊ(π₯π»π§)) β§ ((π₯πΊπ¦)π»π§) = ((π₯π»π§)πΊ(π¦π»π§))) β§ βπ₯ β ran πΊβπ¦ β ran πΊ((π₯π»π¦) = π¦ β§ (π¦π»π₯) = π¦)))) |
6 | 5 | simprrd 773 | . . 3 β’ (π β RingOps β βπ₯ β ran πΊβπ¦ β ran πΊ((π₯π»π¦) = π¦ β§ (π¦π»π₯) = π¦)) |
7 | rngmgmbs4 37404 | . . 3 β’ ((π»:(ran πΊ Γ ran πΊ)βΆran πΊ β§ βπ₯ β ran πΊβπ¦ β ran πΊ((π₯π»π¦) = π¦ β§ (π¦π»π₯) = π¦)) β ran π» = ran πΊ) | |
8 | 4, 6, 7 | syl2anc 583 | . 2 β’ (π β RingOps β ran π» = ran πΊ) |
9 | 8 | eqcomd 2734 | 1 β’ (π β RingOps β ran πΊ = ran π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 βwral 3058 βwrex 3067 Γ cxp 5676 ran crn 5679 βΆwf 6544 βcfv 6548 (class class class)co 7420 1st c1st 7991 2nd c2nd 7992 AbelOpcablo 30367 RingOpscrngo 37367 |
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 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 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 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fo 6554 df-fv 6556 df-ov 7423 df-1st 7993 df-2nd 7994 df-rngo 37368 |
This theorem is referenced by: rngoidmlem 37409 rngo1cl 37412 isdrngo2 37431 |
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