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Theorem grpridd 5639
Description: Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grprinvlem.c ((φ x B y B) → (x + y) B)
grprinvlem.o (φ𝑂 B)
grprinvlem.i ((φ x B) → (𝑂 + x) = x)
grprinvlem.a ((φ (x B y B z B)) → ((x + y) + z) = (x + (y + z)))
grprinvlem.n ((φ x B) → y B (y + x) = 𝑂)
Assertion
Ref Expression
grpridd ((φ x B) → (x + 𝑂) = x)
Distinct variable groups:   x,y,z,B   x,𝑂,y,z   φ,x,y,z   x, + ,y,z

Proof of Theorem grpridd
Dummy variables u 𝑛 v w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.n . . . 4 ((φ x B) → y B (y + x) = 𝑂)
2 oveq1 5462 . . . . . 6 (y = 𝑛 → (y + x) = (𝑛 + x))
32eqeq1d 2045 . . . . 5 (y = 𝑛 → ((y + x) = 𝑂 ↔ (𝑛 + x) = 𝑂))
43cbvrexv 2528 . . . 4 (y B (y + x) = 𝑂𝑛 B (𝑛 + x) = 𝑂)
51, 4sylib 127 . . 3 ((φ x B) → 𝑛 B (𝑛 + x) = 𝑂)
6 grprinvlem.a . . . . . . . 8 ((φ (x B y B z B)) → ((x + y) + z) = (x + (y + z)))
76caovassg 5601 . . . . . . 7 ((φ (u B v B w B)) → ((u + v) + w) = (u + (v + w)))
87adantlr 446 . . . . . 6 (((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) (u B v B w B)) → ((u + v) + w) = (u + (v + w)))
9 simprl 483 . . . . . 6 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → x B)
10 simprrl 491 . . . . . 6 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → 𝑛 B)
118, 9, 10, 9caovassd 5602 . . . . 5 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → ((x + 𝑛) + x) = (x + (𝑛 + x)))
12 grprinvlem.c . . . . . . 7 ((φ x B y B) → (x + y) B)
13 grprinvlem.o . . . . . . 7 (φ𝑂 B)
14 grprinvlem.i . . . . . . 7 ((φ x B) → (𝑂 + x) = x)
15 simprrr 492 . . . . . . 7 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → (𝑛 + x) = 𝑂)
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 5638 . . . . . 6 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → (x + 𝑛) = 𝑂)
1716oveq1d 5470 . . . . 5 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → ((x + 𝑛) + x) = (𝑂 + x))
1815oveq2d 5471 . . . . 5 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → (x + (𝑛 + x)) = (x + 𝑂))
1911, 17, 183eqtr3d 2077 . . . 4 ((φ (x B (𝑛 B (𝑛 + x) = 𝑂))) → (𝑂 + x) = (x + 𝑂))
2019anassrs 380 . . 3 (((φ x B) (𝑛 B (𝑛 + x) = 𝑂)) → (𝑂 + x) = (x + 𝑂))
215, 20rexlimddv 2431 . 2 ((φ x B) → (𝑂 + x) = (x + 𝑂))
2221, 14eqtr3d 2071 1 ((φ x B) → (x + 𝑂) = x)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wrex 2301  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by: (None)
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