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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoaass | Structured version Visualization version GIF version |
Description: The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringgcl.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringgcl.2 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
rngoaass | ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgcl.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 34667 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | ringgcl.2 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 3 | grpoass 28072 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))) |
5 | 2, 4 | sylan 572 | 1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ran crn 5404 ‘cfv 6185 (class class class)co 6974 1st c1st 7497 GrpOpcgr 28058 RingOpscrngo 34651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-fo 6191 df-fv 6193 df-ov 6977 df-1st 7499 df-2nd 7500 df-grpo 28062 df-ablo 28114 df-rngo 34652 |
This theorem is referenced by: (None) |
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