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Theorem caovmo 5645
 Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by set.mm contributors, 4-Mar-1996.)
Hypotheses
Ref Expression
caovmo.1 A V
caovmo.2 B S
caovmo.dom dom F = (S × S)
caovmo.3 ¬ S
caovmo.com (xFy) = (yFx)
caovmo.ass ((xFy)Fz) = (xF(yFz))
caovmo.id (x S → (xFB) = x)
Assertion
Ref Expression
caovmo ∃*w(AFw) = B
Distinct variable groups:   x,y,z,F   x,S,y,z   x,A,y,z   x,B,y,z,w   w,S   w,A   w,B   w,F

Proof of Theorem caovmo
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . . . . 5 (w = v → (w Sv S))
2 oveq2 5531 . . . . . 6 (w = v → (AFw) = (AFv))
32eqeq1d 2361 . . . . 5 (w = v → ((AFw) = B ↔ (AFv) = B))
41, 3anbi12d 691 . . . 4 (w = v → ((w S (AFw) = B) ↔ (v S (AFv) = B)))
54mo4 2237 . . 3 (∃*w(w S (AFw) = B) ↔ wv(((w S (AFw) = B) (v S (AFv) = B)) → w = v))
6 caovmo.1 . . . . . . . . 9 A V
7 vex 2862 . . . . . . . . 9 w V
8 vex 2862 . . . . . . . . 9 v V
9 caovmo.ass . . . . . . . . 9 ((xFy)Fz) = (xF(yFz))
106, 7, 8, 9caovass 5627 . . . . . . . 8 ((AFw)Fv) = (AF(wFv))
11 caovmo.com . . . . . . . . 9 (xFy) = (yFx)
126, 7, 8, 11, 9caov12 5636 . . . . . . . 8 (AF(wFv)) = (wF(AFv))
1310, 12eqtri 2373 . . . . . . 7 ((AFw)Fv) = (wF(AFv))
14 oveq2 5531 . . . . . . . 8 ((AFv) = B → (wF(AFv)) = (wFB))
15 oveq1 5530 . . . . . . . . . 10 (x = w → (xFB) = (wFB))
16 id 19 . . . . . . . . . 10 (x = wx = w)
1715, 16eqeq12d 2367 . . . . . . . . 9 (x = w → ((xFB) = x ↔ (wFB) = w))
18 caovmo.id . . . . . . . . 9 (x S → (xFB) = x)
1917, 18vtoclga 2920 . . . . . . . 8 (w S → (wFB) = w)
2014, 19sylan9eqr 2407 . . . . . . 7 ((w S (AFv) = B) → (wF(AFv)) = w)
2113, 20syl5eq 2397 . . . . . 6 ((w S (AFv) = B) → ((AFw)Fv) = w)
2221ad2ant2rl 729 . . . . 5 (((w S (AFw) = B) (v S (AFv) = B)) → ((AFw)Fv) = w)
23 oveq1 5530 . . . . . . 7 ((AFw) = B → ((AFw)Fv) = (BFv))
24 caovmo.2 . . . . . . . . . 10 B S
2524elexi 2868 . . . . . . . . 9 B V
2625, 8, 11caovcom 5625 . . . . . . . 8 (BFv) = (vFB)
27 oveq1 5530 . . . . . . . . . 10 (x = v → (xFB) = (vFB))
28 id 19 . . . . . . . . . 10 (x = vx = v)
2927, 28eqeq12d 2367 . . . . . . . . 9 (x = v → ((xFB) = x ↔ (vFB) = v))
3029, 18vtoclga 2920 . . . . . . . 8 (v S → (vFB) = v)
3126, 30syl5eq 2397 . . . . . . 7 (v S → (BFv) = v)
3223, 31sylan9eq 2405 . . . . . 6 (((AFw) = B v S) → ((AFw)Fv) = v)
3332ad2ant2lr 728 . . . . 5 (((w S (AFw) = B) (v S (AFv) = B)) → ((AFw)Fv) = v)
3422, 33eqtr3d 2387 . . . 4 (((w S (AFw) = B) (v S (AFv) = B)) → w = v)
3534ax-gen 1546 . . 3 v(((w S (AFw) = B) (v S (AFv) = B)) → w = v)
365, 35mpgbir 1550 . 2 ∃*w(w S (AFw) = B)
37 eleq1 2413 . . . . . 6 ((AFw) = B → ((AFw) SB S))
3824, 37mpbiri 224 . . . . 5 ((AFw) = B → (AFw) S)
39 caovmo.dom . . . . . . 7 dom F = (S × S)
40 caovmo.3 . . . . . . 7 ¬ S
417, 39, 40ndmovrcl 5616 . . . . . 6 ((AFw) S → (A S w S))
4241simprd 449 . . . . 5 ((AFw) Sw S)
4338, 42syl 15 . . . 4 ((AFw) = Bw S)
4443ancri 535 . . 3 ((AFw) = B → (w S (AFw) = B))
4544moimi 2251 . 2 (∃*w(w S (AFw) = B) → ∃*w(AFw) = B)
4636, 45ax-mp 8 1 ∃*w(AFw) = B
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  Vcvv 2859  ∅c0 3550   × cxp 4770  dom cdm 4772  (class class class)co 5525 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fv 4795  df-ov 5526 This theorem is referenced by: (None)
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