MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pj1ghm Structured version   Visualization version   GIF version

Theorem pj1ghm 18316
Description: The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
pj1eu.a + = (+g𝐺)
pj1eu.s = (LSSum‘𝐺)
pj1eu.o 0 = (0g𝐺)
pj1eu.z 𝑍 = (Cntz‘𝐺)
pj1eu.2 (𝜑𝑇 ∈ (SubGrp‘𝐺))
pj1eu.3 (𝜑𝑈 ∈ (SubGrp‘𝐺))
pj1eu.4 (𝜑 → (𝑇𝑈) = { 0 })
pj1eu.5 (𝜑𝑇 ⊆ (𝑍𝑈))
pj1f.p 𝑃 = (proj1𝐺)
Assertion
Ref Expression
pj1ghm (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))

Proof of Theorem pj1ghm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2760 . 2 (Base‘(𝐺s (𝑇 𝑈))) = (Base‘(𝐺s (𝑇 𝑈)))
2 eqid 2760 . 2 (Base‘𝐺) = (Base‘𝐺)
3 ovex 6841 . . 3 (𝑇 𝑈) ∈ V
4 eqid 2760 . . . 4 (𝐺s (𝑇 𝑈)) = (𝐺s (𝑇 𝑈))
5 pj1eu.a . . . 4 + = (+g𝐺)
64, 5ressplusg 16195 . . 3 ((𝑇 𝑈) ∈ V → + = (+g‘(𝐺s (𝑇 𝑈))))
73, 6ax-mp 5 . 2 + = (+g‘(𝐺s (𝑇 𝑈)))
8 pj1eu.2 . . . 4 (𝜑𝑇 ∈ (SubGrp‘𝐺))
9 pj1eu.3 . . . 4 (𝜑𝑈 ∈ (SubGrp‘𝐺))
10 pj1eu.5 . . . 4 (𝜑𝑇 ⊆ (𝑍𝑈))
11 pj1eu.s . . . . 5 = (LSSum‘𝐺)
12 pj1eu.z . . . . 5 𝑍 = (Cntz‘𝐺)
1311, 12lsmsubg 18269 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
148, 9, 10, 13syl3anc 1477 . . 3 (𝜑 → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
154subggrp 17798 . . 3 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝐺s (𝑇 𝑈)) ∈ Grp)
1614, 15syl 17 . 2 (𝜑 → (𝐺s (𝑇 𝑈)) ∈ Grp)
17 subgrcl 17800 . . 3 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
188, 17syl 17 . 2 (𝜑𝐺 ∈ Grp)
19 pj1eu.o . . . . 5 0 = (0g𝐺)
20 pj1eu.4 . . . . 5 (𝜑 → (𝑇𝑈) = { 0 })
21 pj1f.p . . . . 5 𝑃 = (proj1𝐺)
225, 11, 19, 12, 8, 9, 20, 10, 21pj1f 18310 . . . 4 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
232subgss 17796 . . . . 5 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
248, 23syl 17 . . . 4 (𝜑𝑇 ⊆ (Base‘𝐺))
2522, 24fssd 6218 . . 3 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺))
264subgbas 17799 . . . . 5 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2714, 26syl 17 . . . 4 (𝜑 → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2827feq2d 6192 . . 3 (𝜑 → ((𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺) ↔ (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺)))
2925, 28mpbid 222 . 2 (𝜑 → (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺))
3027eleq2d 2825 . . . . 5 (𝜑 → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3127eleq2d 2825 . . . . 5 (𝜑 → (𝑦 ∈ (𝑇 𝑈) ↔ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3230, 31anbi12d 749 . . . 4 (𝜑 → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) ↔ (𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈))))))
3332biimpar 503 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈))))) → (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)))
345, 11, 19, 12, 8, 9, 20, 10, 21pj1id 18312 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑇 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
3534adantrr 755 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
365, 11, 19, 12, 8, 9, 20, 10, 21pj1id 18312 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑇 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3736adantrl 754 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3835, 37oveq12d 6831 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
398adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
40 grpmnd 17630 . . . . . . . 8 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
4139, 17, 403syl 18 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝐺 ∈ Mnd)
4239, 23syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ⊆ (Base‘𝐺))
43 simpl 474 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑥 ∈ (𝑇 𝑈))
44 ffvelrn 6520 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑥 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4522, 43, 44syl2an 495 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4642, 45sseldd 3745 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ (Base‘𝐺))
47 simpr 479 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑦 ∈ (𝑇 𝑈))
48 ffvelrn 6520 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑦 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
4922, 47, 48syl2an 495 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
5042, 49sseldd 3745 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝐺))
519adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
522subgss 17796 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
5351, 52syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈 ⊆ (Base‘𝐺))
545, 11, 19, 12, 8, 9, 20, 10, 21pj2f 18311 . . . . . . . . 9 (𝜑 → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)
55 ffvelrn 6520 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑥 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5654, 43, 55syl2an 495 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5753, 56sseldd 3745 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58 ffvelrn 6520 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑦 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
5954, 47, 58syl2an 495 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
6053, 59sseldd 3745 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝐺))
6110adantr 472 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
6261, 49sseldd 3745 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈))
635, 12cntzi 17962 . . . . . . . 8 ((((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
6462, 56, 63syl2anc 696 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
652, 5, 41, 46, 50, 57, 60, 64mnd4g 17508 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
6638, 65eqtr4d 2797 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
6720adantr 472 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑇𝑈) = { 0 })
685subgcl 17805 . . . . . . . 8 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
69683expb 1114 . . . . . . 7 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
7014, 69sylan 489 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
715subgcl 17805 . . . . . . 7 ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
7239, 45, 49, 71syl3anc 1477 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
735subgcl 17805 . . . . . . 7 ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
7451, 56, 59, 73syl3anc 1477 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
755, 11, 19, 12, 39, 51, 67, 61, 21, 70, 72, 74pj1eq 18313 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))) ↔ (((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥 + 𝑦)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)))))
7666, 75mpbid 222 . . . 4 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥 + 𝑦)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
7776simpld 477 . . 3 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
7833, 77syldan 488 . 2 ((𝜑 ∧ (𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈))))) → ((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
791, 2, 7, 5, 16, 18, 29, 78isghmd 17870 1 (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  wss 3715  {csn 4321  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  s cress 16060  +gcplusg 16143  0gc0g 16302  Mndcmnd 17495  Grpcgrp 17623  SubGrpcsubg 17789   GrpHom cghm 17858  Cntzccntz 17948  LSSumclsm 18249  proj1cpj1 18250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792  df-ghm 17859  df-cntz 17950  df-lsm 18251  df-pj1 18252
This theorem is referenced by:  pj1ghm2  18317  dpjghm  18662  pj1lmhm  19302
  Copyright terms: Public domain W3C validator