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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 36415 | . . 3 β’ (π β RingOps β πΊ β GrpOp) |
3 | eqid 2733 | . . . 4 β’ ran πΊ = ran πΊ | |
4 | 3 | grpofo 29483 | . . 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 36405 | . 2 β’ (π β RingOps β π»:(ran πΊ Γ ran πΊ)βΆran πΊ) |
8 | fof 6757 | . . . 4 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β πΊ:(ran πΊ Γ ran πΊ)βΆran πΊ) | |
9 | 8 | fdmd 6680 | . . 3 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β dom πΊ = (ran πΊ Γ ran πΊ)) |
10 | fdm 6678 | . . . 4 β’ (π»:(ran πΊ Γ ran πΊ)βΆran πΊ β dom π» = (ran πΊ Γ ran πΊ)) | |
11 | eqtr 2756 | . . . . . . 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 5862 | . . . . . 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 415 | . . . . 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 2741 | . . . 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 397 = wceq 1542 β wcel 2107 Γ cxp 5632 dom cdm 5634 ran crn 5635 βΆwf 6493 βontoβwfo 6495 βcfv 6497 1st c1st 7920 2nd c2nd 7921 GrpOpcgr 29473 RingOpscrngo 36399 |
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 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fo 6503 df-fv 6505 df-ov 7361 df-1st 7922 df-2nd 7923 df-grpo 29477 df-ablo 29529 df-rngo 36400 |
This theorem is referenced by: rngorn1 36438 |
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