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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 2733 | . . . 4 β’ ran πΊ = ran πΊ | |
4 | 1, 2, 3 | rngosm 36409 | . . 3 β’ (π β RingOps β π»:(ran πΊ Γ ran πΊ)βΆran πΊ) |
5 | 1, 2, 3 | rngoi 36408 | . . . 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 36440 | . . 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 585 | . 2 β’ (π β RingOps β ran π» = ran πΊ) |
9 | 8 | eqcomd 2739 | 1 β’ (π β RingOps β ran πΊ = ran π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 Γ cxp 5635 ran crn 5638 βΆwf 6496 βcfv 6500 (class class class)co 7361 1st c1st 7923 2nd c2nd 7924 AbelOpcablo 29535 RingOpscrngo 36403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-ov 7364 df-1st 7925 df-2nd 7926 df-rngo 36404 |
This theorem is referenced by: rngoidmlem 36445 rngo1cl 36448 isdrngo2 36467 |
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