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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 18116 | . 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 }) |
6 | 1, 2 | mndidcl 18160 | . . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
7 | 1, 3, 2 | mndlrid 18164 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
8 | 7 | ralrimiva 3098 | . . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)) |
9 | oveq1 7209 | . . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦)) | |
10 | 9 | eqeq1d 2736 | . . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦)) |
11 | 10 | ovanraleqv 7226 | . . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
12 | 11, 4 | elrab2 3598 | . . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))) |
13 | 6, 8, 12 | sylanbrc 586 | . . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂) |
14 | 13 | snssd 4712 | . 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂) |
15 | 5, 14 | eqssd 3908 | 1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 {crab 3058 {csn 4531 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 +gcplusg 16767 0gc0g 16916 Mndcmnd 18145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-iota 6327 df-fun 6371 df-fv 6377 df-riota 7159 df-ov 7205 df-0g 16918 df-mgm 18086 df-sgrp 18135 df-mnd 18146 |
This theorem is referenced by: gsumz 18234 gsumval3a 19260 |
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