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Mirrors > Home > MPE Home > Th. List > gsumvallem2 | Structured version Visualization version GIF version |
Description: Lemma for properties of the set of identities of 𝐺. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
gsumvallem2.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumvallem2.z | ⊢ 0 = (0g‘𝐺) |
gsumvallem2.p | ⊢ + = (+g‘𝐺) |
gsumvallem2.o | ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} |
Ref | Expression |
---|---|
gsumvallem2 | ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumvallem2.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumvallem2.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsumvallem2.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | gsumvallem2.o | . . 3 ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} | |
5 | 1, 2, 3, 4 | mgmidsssn0 17316 | . 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 }) |
6 | 1, 2 | mndidcl 17355 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
7 | 1, 3, 2 | mndlrid 17357 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
8 | 7 | ralrimiva 2995 | . . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
9 | oveq1 6697 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦)) | |
10 | 9 | eqeq1d 2653 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦)) |
11 | oveq2 6698 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑦 + 𝑥) = (𝑦 + 0 )) | |
12 | 11 | eqeq1d 2653 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑦 + 𝑥) = 𝑦 ↔ (𝑦 + 0 ) = 𝑦)) |
13 | 10, 12 | anbi12d 747 | . . . . . 6 ⊢ (𝑥 = 0 → (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
14 | 13 | ralbidv 3015 | . . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
15 | 14, 4 | elrab2 3399 | . . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
16 | 6, 8, 15 | sylanbrc 699 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂) |
17 | 16 | snssd 4372 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂) |
18 | 5, 17 | eqssd 3653 | 1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 {crab 2945 {csn 4210 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 0gc0g 16147 Mndcmnd 17341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-riota 6651 df-ov 6693 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 |
This theorem is referenced by: gsumz 17421 gsumval3a 18350 |
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