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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 17092 | . 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 }) |
6 | 1, 2 | mndidcl 17131 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
7 | 1, 3, 2 | mndlrid 17133 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
8 | 7 | ralrimiva 2949 | . . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
9 | oveq1 6556 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦)) | |
10 | 9 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦)) |
11 | oveq2 6557 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝑦 + 𝑥) = (𝑦 + 0 )) | |
12 | 11 | eqeq1d 2612 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝑦 + 𝑥) = 𝑦 ↔ (𝑦 + 0 ) = 𝑦)) |
13 | 10, 12 | anbi12d 743 | . . . . . 6 ⊢ (𝑥 = 0 → (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
14 | 13 | ralbidv 2969 | . . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
15 | 14, 4 | elrab2 3333 | . . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
16 | 6, 8, 15 | sylanbrc 695 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂) |
17 | 16 | snssd 4281 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂) |
18 | 5, 17 | eqssd 3585 | 1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 {csn 4125 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Mndcmnd 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 |
This theorem is referenced by: gsumz 17197 gsumval3a 18127 |
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