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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngodm1dm2 | Structured version Visualization version GIF version |
Description: In a unital ring the domain of the first variable 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 | β’ π» = (2nd βπ ) |
rnplrnml0.2 | β’ πΊ = (1st βπ ) |
Ref | Expression |
---|---|
rngodm1dm2 | β’ (π β RingOps β dom dom πΊ = dom dom π») |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnplrnml0.2 | . . . 4 β’ πΊ = (1st βπ ) | |
2 | 1 | rngogrpo 37436 | . . 3 β’ (π β RingOps β πΊ β GrpOp) |
3 | eqid 2725 | . . . 4 β’ ran πΊ = ran πΊ | |
4 | 3 | grpofo 30348 | . . 3 β’ (πΊ β GrpOp β πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ) |
5 | 2, 4 | syl 17 | . 2 β’ (π β RingOps β πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ) |
6 | rnplrnml0.1 | . . 3 β’ π» = (2nd βπ ) | |
7 | 1, 6, 3 | rngosm 37426 | . 2 β’ (π β RingOps β π»:(ran πΊ Γ ran πΊ)βΆran πΊ) |
8 | fof 6804 | . . . 4 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β πΊ:(ran πΊ Γ ran πΊ)βΆran πΊ) | |
9 | 8 | fdmd 6727 | . . 3 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β dom πΊ = (ran πΊ Γ ran πΊ)) |
10 | fdm 6726 | . . . 4 β’ (π»:(ran πΊ Γ ran πΊ)βΆran πΊ β dom π» = (ran πΊ Γ ran πΊ)) | |
11 | eqtr 2748 | . . . . . . 7 β’ ((dom πΊ = (ran πΊ Γ ran πΊ) β§ (ran πΊ Γ ran πΊ) = dom π») β dom πΊ = dom π») | |
12 | 11 | dmeqd 5903 | . . . . . 6 β’ ((dom πΊ = (ran πΊ Γ ran πΊ) β§ (ran πΊ Γ ran πΊ) = dom π») β dom dom πΊ = dom dom π») |
13 | 12 | expcom 412 | . . . . 5 β’ ((ran πΊ Γ ran πΊ) = dom π» β (dom πΊ = (ran πΊ Γ ran πΊ) β dom dom πΊ = dom dom π»)) |
14 | 13 | eqcoms 2733 | . . . 4 β’ (dom π» = (ran πΊ Γ ran πΊ) β (dom πΊ = (ran πΊ Γ ran πΊ) β dom dom πΊ = dom dom π»)) |
15 | 10, 14 | syl 17 | . . 3 β’ (π»:(ran πΊ Γ ran πΊ)βΆran πΊ β (dom πΊ = (ran πΊ Γ ran πΊ) β dom dom πΊ = dom dom π»)) |
16 | 9, 15 | syl5com 31 | . 2 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β (π»:(ran πΊ Γ ran πΊ)βΆran πΊ β dom dom πΊ = dom dom π»)) |
17 | 5, 7, 16 | sylc 65 | 1 β’ (π β RingOps β dom dom πΊ = dom dom π») |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Γ cxp 5671 dom cdm 5673 ran crn 5674 βΆwf 6539 βontoβwfo 6541 βcfv 6543 1st c1st 7985 2nd c2nd 7986 GrpOpcgr 30338 RingOpscrngo 37420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-ov 7416 df-1st 7987 df-2nd 7988 df-grpo 30342 df-ablo 30394 df-rngo 37421 |
This theorem is referenced by: rngorn1 37459 |
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