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Theorem pj1ghm 18556
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 2795 . 2 (Base‘(𝐺s (𝑇 𝑈))) = (Base‘(𝐺s (𝑇 𝑈)))
2 eqid 2795 . 2 (Base‘𝐺) = (Base‘𝐺)
3 ovex 7048 . . 3 (𝑇 𝑈) ∈ V
4 eqid 2795 . . . 4 (𝐺s (𝑇 𝑈)) = (𝐺s (𝑇 𝑈))
5 pj1eu.a . . . 4 + = (+g𝐺)
64, 5ressplusg 16441 . . 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 18509 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
148, 9, 10, 13syl3anc 1364 . . 3 (𝜑 → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
154subggrp 18036 . . 3 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝐺s (𝑇 𝑈)) ∈ Grp)
1614, 15syl 17 . 2 (𝜑 → (𝐺s (𝑇 𝑈)) ∈ Grp)
17 subgrcl 18038 . . 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 18550 . . . 4 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
232subgss 18034 . . . . 5 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
248, 23syl 17 . . . 4 (𝜑𝑇 ⊆ (Base‘𝐺))
2522, 24fssd 6396 . . 3 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺))
264subgbas 18037 . . . . 5 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2714, 26syl 17 . . . 4 (𝜑 → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2827feq2d 6368 . . 3 (𝜑 → ((𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺) ↔ (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺)))
2925, 28mpbid 233 . 2 (𝜑 → (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺))
3027eleq2d 2868 . . . . 5 (𝜑 → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3127eleq2d 2868 . . . . 5 (𝜑 → (𝑦 ∈ (𝑇 𝑈) ↔ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3230, 31anbi12d 630 . . . 4 (𝜑 → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) ↔ (𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈))))))
3332biimpar 478 . . 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 18552 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑇 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
3534adantrr 713 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
365, 11, 19, 12, 8, 9, 20, 10, 21pj1id 18552 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑇 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3736adantrl 712 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3835, 37oveq12d 7034 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
398adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
40 grpmnd 17868 . . . . . . . 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 483 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑥 ∈ (𝑇 𝑈))
44 ffvelrn 6714 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑥 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4522, 43, 44syl2an 595 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4642, 45sseldd 3890 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ (Base‘𝐺))
47 simpr 485 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑦 ∈ (𝑇 𝑈))
48 ffvelrn 6714 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑦 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
4922, 47, 48syl2an 595 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
5042, 49sseldd 3890 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝐺))
519adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
522subgss 18034 . . . . . . . . 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 18551 . . . . . . . . 9 (𝜑 → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)
55 ffvelrn 6714 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑥 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5654, 43, 55syl2an 595 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5753, 56sseldd 3890 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58 ffvelrn 6714 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑦 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
5954, 47, 58syl2an 595 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
6053, 59sseldd 3890 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝐺))
6110adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
6261, 49sseldd 3890 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈))
635, 12cntzi 18200 . . . . . . . 8 ((((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
6462, 56, 63syl2anc 584 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
652, 5, 41, 46, 50, 57, 60, 64mnd4g 17746 . . . . . 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 2834 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
6720adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑇𝑈) = { 0 })
685subgcl 18043 . . . . . . . 8 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
69683expb 1113 . . . . . . 7 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
7014, 69sylan 580 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
715subgcl 18043 . . . . . . 7 ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
7239, 45, 49, 71syl3anc 1364 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
735subgcl 18043 . . . . . . 7 ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
7451, 56, 59, 73syl3anc 1364 . . . . . 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 18553 . . . . 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 233 . . . 4 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥 + 𝑦)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
7776simpld 495 . . 3 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
7833, 77syldan 591 . 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 18108 1 (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  cin 3858  wss 3859  {csn 4472  wf 6221  cfv 6225  (class class class)co 7016  Basecbs 16312  s cress 16313  +gcplusg 16394  0gc0g 16542  Mndcmnd 17733  Grpcgrp 17861  SubGrpcsubg 18027   GrpHom cghm 18096  Cntzccntz 18186  LSSumclsm 18489  proj1cpj1 18490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-subg 18030  df-ghm 18097  df-cntz 18188  df-lsm 18491  df-pj1 18492
This theorem is referenced by:  pj1ghm2  18557  dpjghm  18902  pj1lmhm  19562
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