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Theorem nsgqusf1olem2 32965
Description: Lemma for nsgqusf1o 32967. (Contributed by Thierry Arnoux, 4-Aug-2024.)
Hypotheses
Ref Expression
nsgqusf1o.b 𝐵 = (Base‘𝐺)
nsgqusf1o.s 𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}
nsgqusf1o.t 𝑇 = (SubGrp‘𝑄)
nsgqusf1o.1 = (le‘(toInc‘𝑆))
nsgqusf1o.2 = (le‘(toInc‘𝑇))
nsgqusf1o.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
nsgqusf1o.p = (LSSum‘𝐺)
nsgqusf1o.e 𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))
nsgqusf1o.f 𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
nsgqusf1o.n (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
Assertion
Ref Expression
nsgqusf1olem2 (𝜑 → ran 𝐸 = 𝑇)
Distinct variable groups:   ,𝑎,𝑓,,𝑥   𝐵,𝑎,𝑓,,𝑥   𝐸,𝑎,𝑓,,𝑥   𝑓,𝐹,,𝑥   𝐺,𝑎,𝑓,,𝑥   𝑁,𝑎,𝑓,,𝑥   𝑄,𝑎,𝑓,,𝑥   𝑆,𝑎,𝑓,,𝑥   𝑇,𝑎,𝑓,,𝑥   𝜑,𝑎,𝑓,,𝑥
Allowed substitution hints:   𝐹(𝑎)   (𝑥,𝑓,,𝑎)   (𝑥,𝑓,,𝑎)

Proof of Theorem nsgqusf1olem2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
2 nsgqusf1o.s . . . . . . . . . 10 𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}
32reqabi 3453 . . . . . . . . 9 (𝑆 ↔ ( ∈ (SubGrp‘𝐺) ∧ 𝑁))
4 nsgqusf1o.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
5 nsgqusf1o.t . . . . . . . . . . . 12 𝑇 = (SubGrp‘𝑄)
6 nsgqusf1o.1 . . . . . . . . . . . 12 = (le‘(toInc‘𝑆))
7 nsgqusf1o.2 . . . . . . . . . . . 12 = (le‘(toInc‘𝑇))
8 nsgqusf1o.q . . . . . . . . . . . 12 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
9 nsgqusf1o.p . . . . . . . . . . . 12 = (LSSum‘𝐺)
10 nsgqusf1o.e . . . . . . . . . . . 12 𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))
11 nsgqusf1o.f . . . . . . . . . . . 12 𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
12 nsgqusf1o.n . . . . . . . . . . . 12 (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
134, 2, 5, 6, 7, 8, 9, 10, 11, 12nsgqusf1olem1 32964 . . . . . . . . . . 11 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1413anasss 466 . . . . . . . . . 10 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
1514, 5eleqtrdi 2842 . . . . . . . . 9 ((𝜑 ∧ ( ∈ (SubGrp‘𝐺) ∧ 𝑁)) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
163, 15sylan2b 593 . . . . . . . 8 ((𝜑𝑆) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
1716adantr 480 . . . . . . 7 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
181, 17eqeltrd 2832 . . . . . 6 (((𝜑𝑆) ∧ 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
1918r19.29an 3157 . . . . 5 ((𝜑 ∧ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))) → 𝑓 ∈ (SubGrp‘𝑄))
20 sseq2 4008 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑁𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}))
2112adantr 480 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
22 simpr 484 . . . . . . . . 9 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ∈ (SubGrp‘𝑄))
234, 8, 9, 21, 22nsgmgclem 32962 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ (SubGrp‘𝐺))
245eleq2i 2824 . . . . . . . . 9 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
25 nsgsubg 19081 . . . . . . . . . . . . 13 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
2612, 25syl 17 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (SubGrp‘𝐺))
274subgss 19050 . . . . . . . . . . . 12 (𝑁 ∈ (SubGrp‘𝐺) → 𝑁𝐵)
2826, 27syl 17 . . . . . . . . . . 11 (𝜑𝑁𝐵)
2928adantr 480 . . . . . . . . . 10 ((𝜑𝑓𝑇) → 𝑁𝐵)
3026ad2antrr 723 . . . . . . . . . . . 12 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁 ∈ (SubGrp‘𝐺))
319grplsmid 32954 . . . . . . . . . . . 12 ((𝑁 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3230, 31sylancom 587 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) = 𝑁)
3324biimpi 215 . . . . . . . . . . . . 13 (𝑓𝑇𝑓 ∈ (SubGrp‘𝑄))
348nsgqus0 32961 . . . . . . . . . . . . 13 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑓 ∈ (SubGrp‘𝑄)) → 𝑁𝑓)
3512, 33, 34syl2an 595 . . . . . . . . . . . 12 ((𝜑𝑓𝑇) → 𝑁𝑓)
3635adantr 480 . . . . . . . . . . 11 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → 𝑁𝑓)
3732, 36eqeltrd 2832 . . . . . . . . . 10 (((𝜑𝑓𝑇) ∧ 𝑎𝑁) → ({𝑎} 𝑁) ∈ 𝑓)
3829, 37ssrabdv 4071 . . . . . . . . 9 ((𝜑𝑓𝑇) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
3924, 38sylan2br 594 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑁 ⊆ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
4020, 23, 39elrabd 3685 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ { ∈ (SubGrp‘𝐺) ∣ 𝑁})
4140, 2eleqtrrdi 2843 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∈ 𝑆)
42 mpteq1 5241 . . . . . . . . 9 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑥 ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4342rneqd 5937 . . . . . . . 8 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → ran (𝑥 ↦ ({𝑥} 𝑁)) = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
4443eqeq2d 2742 . . . . . . 7 ( = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
4544adantl 481 . . . . . 6 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ = {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) → (𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
46 eqid 2731 . . . . . . . . . . . . . . 15 (Base‘𝑄) = (Base‘𝑄)
4746subgss 19050 . . . . . . . . . . . . . 14 (𝑓 ∈ (SubGrp‘𝑄) → 𝑓 ⊆ (Base‘𝑄))
4847adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 ⊆ (Base‘𝑄))
4948sselda 3982 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑖 ∈ (Base‘𝑄))
508a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
514a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝐵 = (Base‘𝐺))
52 ovexd 7447 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐺 ~QG 𝑁) ∈ V)
53 subgrcl 19054 . . . . . . . . . . . . . . 15 (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
5426, 53syl 17 . . . . . . . . . . . . . 14 (𝜑𝐺 ∈ Grp)
5554ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝐺 ∈ Grp)
5650, 51, 52, 55qusbas 17498 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
5749, 56eleqtrrd 2835 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → 𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)))
58 elqsi 8770 . . . . . . . . . . 11 (𝑖 ∈ (𝐵 / (𝐺 ~QG 𝑁)) → ∃𝑥𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
5957, 58syl 17 . . . . . . . . . 10 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ∃𝑥𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁))
60 sneq 4638 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → {𝑎} = {𝑥})
6160oveq1d 7427 . . . . . . . . . . . . . . 15 (𝑎 = 𝑥 → ({𝑎} 𝑁) = ({𝑥} 𝑁))
6261eleq1d 2817 . . . . . . . . . . . . . 14 (𝑎 = 𝑥 → (({𝑎} 𝑁) ∈ 𝑓 ↔ ({𝑥} 𝑁) ∈ 𝑓))
63 simplr 766 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥𝐵)
64 simpr 484 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
6526ad4antr 729 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
664, 9, 65, 63quslsm 32956 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
6764, 66eqtrd 2771 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖 = ({𝑥} 𝑁))
68 simpllr 773 . . . . . . . . . . . . . . 15 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑖𝑓)
6967, 68eqeltrrd 2833 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → ({𝑥} 𝑁) ∈ 𝑓)
7062, 63, 69elrabd 3685 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7170, 67jca 511 . . . . . . . . . . . 12 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) ∧ 𝑥𝐵) ∧ 𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁)))
7271expl 457 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ((𝑥𝐵𝑖 = [𝑥](𝐺 ~QG 𝑁)) → (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ∧ 𝑖 = ({𝑥} 𝑁))))
7372reximdv2 3163 . . . . . . . . . 10 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → (∃𝑥𝐵 𝑖 = [𝑥](𝐺 ~QG 𝑁) → ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
7459, 73mpd 15 . . . . . . . . 9 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖𝑓) → ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁))
75 simplr 766 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
7662elrab 3683 . . . . . . . . . . . 12 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↔ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
7775, 76sylib 217 . . . . . . . . . . 11 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓))
78 simpllr 773 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖 = ({𝑥} 𝑁))
79 simpr 484 . . . . . . . . . . . . . 14 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → ({𝑥} 𝑁) ∈ 𝑓)
8078, 79eqeltrd 2832 . . . . . . . . . . . . 13 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑥𝐵) ∧ ({𝑥} 𝑁) ∈ 𝑓) → 𝑖𝑓)
8180anasss 466 . . . . . . . . . . . 12 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8281adantllr 716 . . . . . . . . . . 11 (((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ (𝑥𝐵 ∧ ({𝑥} 𝑁) ∈ 𝑓)) → 𝑖𝑓)
8377, 82mpdan 684 . . . . . . . . . 10 ((((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ 𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8483r19.29an 3157 . . . . . . . . 9 (((𝜑𝑓 ∈ (SubGrp‘𝑄)) ∧ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)) → 𝑖𝑓)
8574, 84impbida 798 . . . . . . . 8 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓 ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
86 eqid 2731 . . . . . . . . . 10 (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) = (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))
8786elrnmpt 5955 . . . . . . . . 9 (𝑖 ∈ V → (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁)))
8887elv 3479 . . . . . . . 8 (𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓}𝑖 = ({𝑥} 𝑁))
8985, 88bitr4di 289 . . . . . . 7 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → (𝑖𝑓𝑖 ∈ ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁))))
9089eqrdv 2729 . . . . . 6 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → 𝑓 = ran (𝑥 ∈ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓} ↦ ({𝑥} 𝑁)))
9141, 45, 90rspcedvd 3614 . . . . 5 ((𝜑𝑓 ∈ (SubGrp‘𝑄)) → ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)))
9219, 91impbida 798 . . . 4 (𝜑 → (∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ 𝑓 ∈ (SubGrp‘𝑄)))
9392abbidv 2800 . . 3 (𝜑 → {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))} = {𝑓𝑓 ∈ (SubGrp‘𝑄)})
9410rnmpt 5954 . . 3 ran 𝐸 = {𝑓 ∣ ∃𝑆 𝑓 = ran (𝑥 ↦ ({𝑥} 𝑁))}
95 abid1 2869 . . 3 (SubGrp‘𝑄) = {𝑓𝑓 ∈ (SubGrp‘𝑄)}
9693, 94, 953eqtr4g 2796 . 2 (𝜑 → ran 𝐸 = (SubGrp‘𝑄))
9796, 5eqtr4di 2789 1 (𝜑 → ran 𝐸 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  {cab 2708  wrex 3069  {crab 3431  Vcvv 3473  wss 3948  {csn 4628  cmpt 5231  ran crn 5677  cfv 6543  (class class class)co 7412  [cec 8707   / cqs 8708  Basecbs 17151  lecple 17211   /s cqus 17458  toInccipo 18490  Grpcgrp 18861  SubGrpcsubg 19043  NrmSGrpcnsg 19044   ~QG cqg 19045  LSSumclsm 19550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-tpos 8217  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-ec 8711  df-qs 8715  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12480  df-z 12566  df-dec 12685  df-uz 12830  df-fz 13492  df-struct 17087  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-sca 17220  df-vsca 17221  df-ip 17222  df-tset 17223  df-ple 17224  df-ds 17226  df-0g 17394  df-imas 17461  df-qus 17462  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-submnd 18712  df-grp 18864  df-minusg 18865  df-subg 19046  df-nsg 19047  df-eqg 19048  df-oppg 19258  df-lsm 19552
This theorem is referenced by:  nsgqusf1o  32967
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