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Theorem mndvrid 21042
 Description: Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
mndvcl.b 𝐵 = (Base‘𝑀)
mndvcl.p + = (+g𝑀)
mndvlid.z 0 = (0g𝑀)
Assertion
Ref Expression
mndvrid ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) → (𝑋f + (𝐼 × { 0 })) = 𝑋)

Proof of Theorem mndvrid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elmapex 8428 . . . 4 (𝑋 ∈ (𝐵m 𝐼) → (𝐵 ∈ V ∧ 𝐼 ∈ V))
21simprd 499 . . 3 (𝑋 ∈ (𝐵m 𝐼) → 𝐼 ∈ V)
32adantl 485 . 2 ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) → 𝐼 ∈ V)
4 elmapi 8429 . . 3 (𝑋 ∈ (𝐵m 𝐼) → 𝑋:𝐼𝐵)
54adantl 485 . 2 ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) → 𝑋:𝐼𝐵)
6 mndvcl.b . . . 4 𝐵 = (Base‘𝑀)
7 mndvlid.z . . . 4 0 = (0g𝑀)
86, 7mndidcl 17938 . . 3 (𝑀 ∈ Mnd → 0𝐵)
98adantr 484 . 2 ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) → 0𝐵)
10 mndvcl.p . . . 4 + = (+g𝑀)
116, 10, 7mndrid 17944 . . 3 ((𝑀 ∈ Mnd ∧ 𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
1211adantlr 714 . 2 (((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) ∧ 𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
133, 5, 9, 12caofid0r 7431 1 ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) → (𝑋f + (𝐼 × { 0 })) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3442  {csn 4528   × cxp 5521  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145   ∘f cof 7398   ↑m cmap 8407  Basecbs 16495  +gcplusg 16577  0gc0g 16725  Mndcmnd 17923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7400  df-1st 7684  df-2nd 7685  df-map 8409  df-0g 16727  df-mgm 17864  df-sgrp 17913  df-mnd 17924 This theorem is referenced by: (None)
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