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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 18686 | . 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 }) | 
| 6 | 1, 2 | mndidcl 18763 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) | 
| 7 | 1, 3, 2 | mndlrid 18767 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) | 
| 8 | 7 | ralrimiva 3145 | . . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) | 
| 9 | oveq1 7439 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦)) | |
| 10 | 9 | eqeq1d 2738 | . . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦)) | 
| 11 | 10 | ovanraleqv 7456 | . . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) | 
| 12 | 11, 4 | elrab2 3694 | . . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) | 
| 13 | 6, 8, 12 | sylanbrc 583 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂) | 
| 14 | 13 | snssd 4808 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂) | 
| 15 | 5, 14 | eqssd 4000 | 1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 }) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 {crab 3435 {csn 4625 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Mndcmnd 18748 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 | 
| This theorem is referenced by: gsumz 18850 gsumval3a 19922 | 
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