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Theorem pj1ghm 19670
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 2725 . 2 (Base‘(𝐺s (𝑇 𝑈))) = (Base‘(𝐺s (𝑇 𝑈)))
2 eqid 2725 . 2 (Base‘𝐺) = (Base‘𝐺)
3 ovex 7452 . . 3 (𝑇 𝑈) ∈ V
4 eqid 2725 . . . 4 (𝐺s (𝑇 𝑈)) = (𝐺s (𝑇 𝑈))
5 pj1eu.a . . . 4 + = (+g𝐺)
64, 5ressplusg 17274 . . 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 19621 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
148, 9, 10, 13syl3anc 1368 . . 3 (𝜑 → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
154subggrp 19092 . . 3 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝐺s (𝑇 𝑈)) ∈ Grp)
1614, 15syl 17 . 2 (𝜑 → (𝐺s (𝑇 𝑈)) ∈ Grp)
17 subgrcl 19094 . . 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 19664 . . . 4 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
232subgss 19090 . . . . 5 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
248, 23syl 17 . . . 4 (𝜑𝑇 ⊆ (Base‘𝐺))
2522, 24fssd 6740 . . 3 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺))
264subgbas 19093 . . . . 5 ((𝑇 𝑈) ∈ (SubGrp‘𝐺) → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2714, 26syl 17 . . . 4 (𝜑 → (𝑇 𝑈) = (Base‘(𝐺s (𝑇 𝑈))))
2827feq2d 6709 . . 3 (𝜑 → ((𝑇𝑃𝑈):(𝑇 𝑈)⟶(Base‘𝐺) ↔ (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺)))
2925, 28mpbid 231 . 2 (𝜑 → (𝑇𝑃𝑈):(Base‘(𝐺s (𝑇 𝑈)))⟶(Base‘𝐺))
3027eleq2d 2811 . . . . 5 (𝜑 → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (Base‘(𝐺s (𝑇 𝑈)))))
3127eleq2d 2811 . . . . 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 476 . . 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 19666 . . . . . . . 8 ((𝜑𝑥 ∈ (𝑇 𝑈)) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
3534adantrr 715 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑥 = (((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)))
365, 11, 19, 12, 8, 9, 20, 10, 21pj1id 19666 . . . . . . . 8 ((𝜑𝑦 ∈ (𝑇 𝑈)) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3736adantrl 714 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑦 = (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦)))
3835, 37oveq12d 7437 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑈𝑃𝑇)‘𝑥)) + (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑦))))
398adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
40 grpmnd 18905 . . . . . . . 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 481 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑥 ∈ (𝑇 𝑈))
44 ffvelcdm 7090 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑥 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4522, 43, 44syl2an 594 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇)
4642, 45sseldd 3977 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑥) ∈ (Base‘𝐺))
47 simpr 483 . . . . . . . . 9 ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → 𝑦 ∈ (𝑇 𝑈))
48 ffvelcdm 7090 . . . . . . . . 9 (((𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇𝑦 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
4922, 47, 48syl2an 594 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇)
5042, 49sseldd 3977 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (Base‘𝐺))
519adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
522subgss 19090 . . . . . . . . 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 19665 . . . . . . . . 9 (𝜑 → (𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈)
55 ffvelcdm 7090 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑥 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5654, 43, 55syl2an 594 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈)
5753, 56sseldd 3977 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑥) ∈ (Base‘𝐺))
58 ffvelcdm 7090 . . . . . . . . 9 (((𝑈𝑃𝑇):(𝑇 𝑈)⟶𝑈𝑦 ∈ (𝑇 𝑈)) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
5954, 47, 58syl2an 594 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈)
6053, 59sseldd 3977 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑈𝑃𝑇)‘𝑦) ∈ (Base‘𝐺))
6110adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
6261, 49sseldd 3977 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈))
635, 12cntzi 19292 . . . . . . . 8 ((((𝑇𝑃𝑈)‘𝑦) ∈ (𝑍𝑈) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
6462, 56, 63syl2anc 582 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑦) + ((𝑈𝑃𝑇)‘𝑥)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
652, 5, 41, 46, 50, 57, 60, 64mnd4g 18711 . . . . . 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 2768 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) = ((((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) + (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
6720adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑇𝑈) = { 0 })
685subgcl 19099 . . . . . . . 8 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
69683expb 1117 . . . . . . 7 (((𝑇 𝑈) ∈ (SubGrp‘𝐺) ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
7014, 69sylan 578 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (𝑥 + 𝑦) ∈ (𝑇 𝑈))
715subgcl 19099 . . . . . . 7 ((𝑇 ∈ (SubGrp‘𝐺) ∧ ((𝑇𝑃𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑇𝑃𝑈)‘𝑦) ∈ 𝑇) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
7239, 45, 49, 71syl3anc 1368 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∈ 𝑇)
735subgcl 19099 . . . . . . 7 ((𝑈 ∈ (SubGrp‘𝐺) ∧ ((𝑈𝑃𝑇)‘𝑥) ∈ 𝑈 ∧ ((𝑈𝑃𝑇)‘𝑦) ∈ 𝑈) → (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦)) ∈ 𝑈)
7451, 56, 59, 73syl3anc 1368 . . . . . 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 19667 . . . . 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 231 . . . 4 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → (((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)) ∧ ((𝑈𝑃𝑇)‘(𝑥 + 𝑦)) = (((𝑈𝑃𝑇)‘𝑥) + ((𝑈𝑃𝑇)‘𝑦))))
7776simpld 493 . . 3 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈))) → ((𝑇𝑃𝑈)‘(𝑥 + 𝑦)) = (((𝑇𝑃𝑈)‘𝑥) + ((𝑇𝑃𝑈)‘𝑦)))
7833, 77syldan 589 . 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 19188 1 (𝜑 → (𝑇𝑃𝑈) ∈ ((𝐺s (𝑇 𝑈)) GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  cin 3943  wss 3944  {csn 4630  wf 6545  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  +gcplusg 17236  0gc0g 17424  Mndcmnd 18697  Grpcgrp 18898  SubGrpcsubg 19083   GrpHom cghm 19175  Cntzccntz 19278  LSSumclsm 19601  proj1cpj1 19602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-ghm 19176  df-cntz 19280  df-lsm 19603  df-pj1 19604
This theorem is referenced by:  pj1ghm2  19671  dpjghm  20032  pj1lmhm  20997
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