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Theorem grpidssd 18154
Description: If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then both groups have the same identity element. (Contributed by AV, 15-Mar-2019.)
Hypotheses
Ref Expression
grpidssd.m (𝜑𝑀 ∈ Grp)
grpidssd.s (𝜑𝑆 ∈ Grp)
grpidssd.b 𝐵 = (Base‘𝑆)
grpidssd.c (𝜑𝐵 ⊆ (Base‘𝑀))
grpidssd.o (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
Assertion
Ref Expression
grpidssd (𝜑 → (0g𝑀) = (0g𝑆))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem grpidssd
StepHypRef Expression
1 grpidssd.s . . . . . 6 (𝜑𝑆 ∈ Grp)
2 grpidssd.b . . . . . . 7 𝐵 = (Base‘𝑆)
3 eqid 2820 . . . . . . 7 (0g𝑆) = (0g𝑆)
42, 3grpidcl 18110 . . . . . 6 (𝑆 ∈ Grp → (0g𝑆) ∈ 𝐵)
51, 4syl 17 . . . . 5 (𝜑 → (0g𝑆) ∈ 𝐵)
6 grpidssd.o . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
7 oveq1 7140 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)𝑦))
8 oveq1 7140 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)𝑦))
97, 8eqeq12d 2836 . . . . . 6 (𝑥 = (0g𝑆) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦)))
10 oveq2 7141 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)(0g𝑆)))
11 oveq2 7141 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)(0g𝑆)))
1210, 11eqeq12d 2836 . . . . . 6 (𝑦 = (0g𝑆) → (((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆))))
139, 12rspc2va 3613 . . . . 5 ((((0g𝑆) ∈ 𝐵 ∧ (0g𝑆) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
145, 5, 6, 13syl21anc 835 . . . 4 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
15 eqid 2820 . . . . . 6 (+g𝑆) = (+g𝑆)
162, 15, 3grplid 18112 . . . . 5 ((𝑆 ∈ Grp ∧ (0g𝑆) ∈ 𝐵) → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
171, 4, 16syl2anc2 587 . . . 4 (𝜑 → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
1814, 17eqtrd 2855 . . 3 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆))
19 grpidssd.m . . . 4 (𝜑𝑀 ∈ Grp)
20 grpidssd.c . . . . 5 (𝜑𝐵 ⊆ (Base‘𝑀))
2120, 5sseldd 3947 . . . 4 (𝜑 → (0g𝑆) ∈ (Base‘𝑀))
22 eqid 2820 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
23 eqid 2820 . . . . 5 (+g𝑀) = (+g𝑀)
24 eqid 2820 . . . . 5 (0g𝑀) = (0g𝑀)
2522, 23, 24grpidlcan 18144 . . . 4 ((𝑀 ∈ Grp ∧ (0g𝑆) ∈ (Base‘𝑀) ∧ (0g𝑆) ∈ (Base‘𝑀)) → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2619, 21, 21, 25syl3anc 1367 . . 3 (𝜑 → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2718, 26mpbid 234 . 2 (𝜑 → (0g𝑆) = (0g𝑀))
2827eqcomd 2826 1 (𝜑 → (0g𝑀) = (0g𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1537  wcel 2114  wral 3125  wss 3913  cfv 6331  (class class class)co 7133  Basecbs 16462  +gcplusg 16544  0gc0g 16692  Grpcgrp 18082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-iota 6290  df-fun 6333  df-fv 6339  df-riota 7091  df-ov 7136  df-0g 16694  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085
This theorem is referenced by:  grpinvssd  18155
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