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Theorem pj1ghm 18056
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 2621 . 2 (Base‘(𝐺s (𝑇 𝑈))) = (Base‘(𝐺s (𝑇 𝑈)))
2 eqid 2621 . 2 (Base‘𝐺) = (Base‘𝐺)
3 ovex 6643 . . 3 (𝑇 𝑈) ∈ V
4 eqid 2621 . . . 4 (𝐺s (𝑇 𝑈)) = (𝐺s (𝑇 𝑈))
5 pj1eu.a . . . 4 + = (+g𝐺)
64, 5ressplusg 15933 . . 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 18009 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
148, 9, 10, 13syl3anc 1323 . . 3 (𝜑 → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
154subggrp 17537 . . 3 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝐺s (𝑇 𝑈)) ∈ Grp)
1614, 15syl 17 . 2 (𝜑 → (𝐺s (𝑇 𝑈)) ∈ Grp)
17 subgrcl 17539 . . 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 18050 . . . 4 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
232subgss 17535 . . . . 5 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
248, 23syl 17 . . . 4 (𝜑𝑇 ⊆ (Base‘𝐺))
2522, 24fssd 6024 . . 3 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺))
264subgbas 17538 . . . . 5 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2714, 26syl 17 . . . 4 (𝜑 → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2827feq2d 5998 . . 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 2684 . . . . 5 (𝜑 → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3127eleq2d 2684 . . . . 5 (𝜑 → (𝑦 ∈ (𝑇 𝑈) ↔ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3230, 31anbi12d 746 . . . 4 (𝜑 → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) ↔ (𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈))))))
3332biimpar 502 . . 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 18052 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑇 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
3534adantrr 752 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
365, 11, 19, 12, 8, 9, 20, 10, 21pj1id 18052 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑇 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3736adantrl 751 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3835, 37oveq12d 6633 . . . . . 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 17369 . . . . . . . 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 473 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑥 ∈ (𝑇 𝑈))
44 ffvelrn 6323 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑥 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4522, 43, 44syl2an 494 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4642, 45sseldd 3589 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ (Base‘𝐺))
47 simpr 477 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑦 ∈ (𝑇 𝑈))
48 ffvelrn 6323 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑦 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
4922, 47, 48syl2an 494 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
5042, 49sseldd 3589 . . . . . . 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 17535 . . . . . . . . 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 18051 . . . . . . . . 9 (𝜑 → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)
55 ffvelrn 6323 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑥 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5654, 43, 55syl2an 494 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5753, 56sseldd 3589 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58 ffvelrn 6323 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑦 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
5954, 47, 58syl2an 494 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
6053, 59sseldd 3589 . . . . . . 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 3589 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈))
635, 12cntzi 17702 . . . . . . . 8 ((((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
6462, 56, 63syl2anc 692 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
652, 5, 41, 46, 50, 57, 60, 64mnd4g 17247 . . . . . 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 2658 . . . . 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 17544 . . . . . . . 8 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
69683expb 1263 . . . . . . 7 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
7014, 69sylan 488 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
715subgcl 17544 . . . . . . 7 ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
7239, 45, 49, 71syl3anc 1323 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
735subgcl 17544 . . . . . . 7 ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
7451, 56, 59, 73syl3anc 1323 . . . . . 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 18053 . . . . 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 475 . . 3 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
7833, 77syldan 487 . 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 17609 1 (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cin 3559  wss 3560  {csn 4155  wf 5853  cfv 5857  (class class class)co 6615  Basecbs 15800  s cress 15801  +gcplusg 15881  0gc0g 16040  Mndcmnd 17234  Grpcgrp 17362  SubGrpcsubg 17528   GrpHom cghm 17597  Cntzccntz 17688  LSSumclsm 17989  proj1cpj1 17990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-grp 17365  df-minusg 17366  df-sbg 17367  df-subg 17531  df-ghm 17598  df-cntz 17690  df-lsm 17991  df-pj1 17992
This theorem is referenced by:  pj1ghm2  18057  dpjghm  18402  pj1lmhm  19040
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