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Theorem gsumvallem2 17580
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.)
Hypotheses
Ref Expression
gsumvallem2.b 𝐵 = (Base‘𝐺)
gsumvallem2.z 0 = (0g𝐺)
gsumvallem2.p + = (+g𝐺)
gsumvallem2.o 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
Assertion
Ref Expression
gsumvallem2 (𝐺 ∈ Mnd → 𝑂 = { 0 })
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥, 0 ,𝑦
Allowed substitution hints:   𝑂(𝑥,𝑦)

Proof of Theorem gsumvallem2
StepHypRef Expression
1 gsumvallem2.b . . 3 𝐵 = (Base‘𝐺)
2 gsumvallem2.z . . 3 0 = (0g𝐺)
3 gsumvallem2.p . . 3 + = (+g𝐺)
4 gsumvallem2.o . . 3 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
51, 2, 3, 4mgmidsssn0 17477 . 2 (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
61, 2mndidcl 17516 . . . 4 (𝐺 ∈ Mnd → 0𝐵)
71, 3, 2mndlrid 17518 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
87ralrimiva 3115 . . . 4 (𝐺 ∈ Mnd → ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9 oveq1 6800 . . . . . . . 8 (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
109eqeq1d 2773 . . . . . . 7 (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
11 oveq2 6801 . . . . . . . 8 (𝑥 = 0 → (𝑦 + 𝑥) = (𝑦 + 0 ))
1211eqeq1d 2773 . . . . . . 7 (𝑥 = 0 → ((𝑦 + 𝑥) = 𝑦 ↔ (𝑦 + 0 ) = 𝑦))
1310, 12anbi12d 616 . . . . . 6 (𝑥 = 0 → (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
1413ralbidv 3135 . . . . 5 (𝑥 = 0 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
1514, 4elrab2 3518 . . . 4 ( 0𝑂 ↔ ( 0𝐵 ∧ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
166, 8, 15sylanbrc 572 . . 3 (𝐺 ∈ Mnd → 0𝑂)
1716snssd 4475 . 2 (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
185, 17eqssd 3769 1 (𝐺 ∈ Mnd → 𝑂 = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  {csn 4316  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-riota 6754  df-ov 6796  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503
This theorem is referenced by:  gsumz  17582  gsumval3a  18511
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