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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngosm | Structured version Visualization version GIF version |
Description: Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringi.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringi.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
ringi.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
rngosm | ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringi.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | ringi.2 | . . . 4 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | ringi.3 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | rngoi 34655 | . . 3 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))) |
5 | 4 | simpld 495 | . 2 ⊢ (𝑅 ∈ RingOps → (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋)) |
6 | 5 | simprd 496 | 1 ⊢ (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1078 = wceq 1520 ∈ wcel 2079 ∀wral 3103 ∃wrex 3104 × cxp 5433 ran crn 5436 ⟶wf 6213 ‘cfv 6217 (class class class)co 7007 1st c1st 7534 2nd c2nd 7535 AbelOpcablo 28000 RingOpscrngo 34650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-fv 6225 df-ov 7010 df-1st 7536 df-2nd 7537 df-rngo 34651 |
This theorem is referenced by: rngocl 34657 rngosn3 34680 rngodm1dm2 34688 rngorn1eq 34690 rngomndo 34691 divrngcl 34713 isdrngo2 34714 |
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