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Theorem pj1ghm 19721
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 2737 . 2 (Base‘(𝐺s (𝑇 𝑈))) = (Base‘(𝐺s (𝑇 𝑈)))
2 eqid 2737 . 2 (Base‘𝐺) = (Base‘𝐺)
3 ovex 7464 . . 3 (𝑇 𝑈) ∈ V
4 eqid 2737 . . . 4 (𝐺s (𝑇 𝑈)) = (𝐺s (𝑇 𝑈))
5 pj1eu.a . . . 4 + = (+g𝐺)
64, 5ressplusg 17334 . . 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 19672 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
148, 9, 10, 13syl3anc 1373 . . 3 (𝜑 → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
154subggrp 19147 . . 3 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝐺s (𝑇 𝑈)) ∈ Grp)
1614, 15syl 17 . 2 (𝜑 → (𝐺s (𝑇 𝑈)) ∈ Grp)
17 subgrcl 19149 . . 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 19715 . . . 4 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
232subgss 19145 . . . . 5 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
248, 23syl 17 . . . 4 (𝜑𝑇 ⊆ (Base‘𝐺))
2522, 24fssd 6753 . . 3 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺))
264subgbas 19148 . . . . 5 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2714, 26syl 17 . . . 4 (𝜑 → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2827feq2d 6722 . . 3 (𝜑 → ((𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺) ↔ (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺)))
2925, 28mpbid 232 . 2 (𝜑 → (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺))
3027eleq2d 2827 . . . . 5 (𝜑 → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3127eleq2d 2827 . . . . 5 (𝜑 → (𝑦 ∈ (𝑇 𝑈) ↔ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3230, 31anbi12d 632 . . . 4 (𝜑 → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) ↔ (𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈))) ∧ 𝑦 ∈ (Base‘(𝐺s (𝑇 𝑈))))))
3332biimpar 477 . . 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 19717 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑇 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
3534adantrr 717 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
365, 11, 19, 12, 8, 9, 20, 10, 21pj1id 19717 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑇 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3736adantrl 716 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3835, 37oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
398adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
40 grpmnd 18958 . . . . . . . 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 482 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑥 ∈ (𝑇 𝑈))
44 ffvelcdm 7101 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑥 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4522, 43, 44syl2an 596 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4642, 45sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ (Base‘𝐺))
47 simpr 484 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑦 ∈ (𝑇 𝑈))
48 ffvelcdm 7101 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑦 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
4922, 47, 48syl2an 596 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
5042, 49sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝐺))
519adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
522subgss 19145 . . . . . . . . 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 19716 . . . . . . . . 9 (𝜑 → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)
55 ffvelcdm 7101 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑥 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5654, 43, 55syl2an 596 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5753, 56sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58 ffvelcdm 7101 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑦 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
5954, 47, 58syl2an 596 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
6053, 59sseldd 3984 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝐺))
6110adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
6261, 49sseldd 3984 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈))
635, 12cntzi 19347 . . . . . . . 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 18761 . . . . . 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 2780 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
6720adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑇𝑈) = { 0 })
685subgcl 19154 . . . . . . . 8 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
69683expb 1121 . . . . . . 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 19154 . . . . . . 7 ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
7239, 45, 49, 71syl3anc 1373 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
735subgcl 19154 . . . . . . 7 ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
7451, 56, 59, 73syl3anc 1373 . . . . . 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 19718 . . . . 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 232 . . . 4 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥 + 𝑦)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
7776simpld 494 . . 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 19243 1 (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  {csn 4626  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  +gcplusg 17297  0gc0g 17484  Mndcmnd 18747  Grpcgrp 18951  SubGrpcsubg 19138   GrpHom cghm 19230  Cntzccntz 19333  LSSumclsm 19652  proj1cpj1 19653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-cntz 19335  df-lsm 19654  df-pj1 19655
This theorem is referenced by:  pj1ghm2  19722  dpjghm  20083  pj1lmhm  21099
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