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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 18617 | . 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 }) |
6 | 1, 2 | mndidcl 18694 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
7 | 1, 3, 2 | mndlrid 18698 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
8 | 7 | ralrimiva 3141 | . . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
9 | oveq1 7421 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦)) | |
10 | 9 | eqeq1d 2729 | . . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦)) |
11 | 10 | ovanraleqv 7438 | . . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
12 | 11, 4 | elrab2 3683 | . . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
13 | 6, 8, 12 | sylanbrc 582 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂) |
14 | 13 | snssd 4808 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂) |
15 | 5, 14 | eqssd 3995 | 1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3056 {crab 3427 {csn 4624 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 +gcplusg 17218 0gc0g 17406 Mndcmnd 18679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-riota 7370 df-ov 7417 df-0g 17408 df-mgm 18585 df-sgrp 18664 df-mnd 18680 |
This theorem is referenced by: gsumz 18773 gsumval3a 19842 |
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