Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nsgqusf1olem1 Structured version   Visualization version   GIF version

Theorem nsgqusf1olem1 31119
 Description: Lemma for nsgqusf1o 31122. (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
nsgqusf1olem1 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 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 nsgqusf1olem1
Dummy variables 𝑖 𝑗 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgqusf1o.n . . . . 5 (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
2 nsgqusf1o.q . . . . . 6 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
32qusgrp 18402 . . . . 5 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
41, 3syl 17 . . . 4 (𝜑𝑄 ∈ Grp)
54ad2antrr 725 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → 𝑄 ∈ Grp)
6 nsgqusf1o.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
76subgss 18347 . . . . . . . . 9 ( ∈ (SubGrp‘𝐺) → 𝐵)
87ad2antlr 726 . . . . . . . 8 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → 𝐵)
98sselda 3892 . . . . . . 7 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → 𝑥𝐵)
10 ovex 7183 . . . . . . . 8 (𝐺 ~QG 𝑁) ∈ V
1110ecelqsi 8363 . . . . . . 7 (𝑥𝐵 → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁)))
129, 11syl 17 . . . . . 6 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁)))
13 nsgqusf1o.p . . . . . . 7 = (LSSum‘𝐺)
14 nsgsubg 18377 . . . . . . . . 9 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
151, 14syl 17 . . . . . . . 8 (𝜑𝑁 ∈ (SubGrp‘𝐺))
1615ad3antrrr 729 . . . . . . 7 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → 𝑁 ∈ (SubGrp‘𝐺))
176, 13, 16, 9quslsm 31114 . . . . . 6 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
182a1i 11 . . . . . . . 8 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
196a1i 11 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐺))
20 ovexd 7185 . . . . . . . 8 (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
21 subgrcl 18351 . . . . . . . . 9 (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2215, 21syl 17 . . . . . . . 8 (𝜑𝐺 ∈ Grp)
2318, 19, 20, 22qusbas 16876 . . . . . . 7 (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
2423ad3antrrr 729 . . . . . 6 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
2512, 17, 243eltr3d 2866 . . . . 5 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ({𝑥} 𝑁) ∈ (Base‘𝑄))
2625ralrimiva 3113 . . . 4 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ∀𝑥 ({𝑥} 𝑁) ∈ (Base‘𝑄))
27 eqid 2758 . . . . 5 (𝑥 ↦ ({𝑥} 𝑁)) = (𝑥 ↦ ({𝑥} 𝑁))
2827rnmptss 6877 . . . 4 (∀𝑥 ({𝑥} 𝑁) ∈ (Base‘𝑄) → ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄))
2926, 28syl 17 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄))
30 nfv 1915 . . . 4 𝑥((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁)
31 ovexd 7185 . . . 4 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ({𝑥} 𝑁) ∈ V)
32 eqid 2758 . . . . . . 7 (0g𝐺) = (0g𝐺)
3332subg0cl 18354 . . . . . 6 ( ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ )
3433ne0d 4234 . . . . 5 ( ∈ (SubGrp‘𝐺) → ≠ ∅)
3534ad2antlr 726 . . . 4 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ≠ ∅)
3630, 31, 27, 35rnmptn0 6073 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ≠ ∅)
37 nfmpt1 5130 . . . . . . . 8 𝑥(𝑥 ↦ ({𝑥} 𝑁))
3837nfrn 5793 . . . . . . 7 𝑥ran (𝑥 ↦ ({𝑥} 𝑁))
3938nfel2 2937 . . . . . 6 𝑥 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4030, 39nfan 1900 . . . . 5 𝑥(((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
4138nfel2 2937 . . . . . . 7 𝑥(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4238, 41nfralw 3153 . . . . . 6 𝑥𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4338nfel2 2937 . . . . . 6 𝑥((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4442, 43nfan 1900 . . . . 5 𝑥(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
45 sneq 4532 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → {𝑥} = {𝑧})
4645oveq1d 7165 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ({𝑥} 𝑁) = ({𝑧} 𝑁))
4746cbvmptv 5135 . . . . . . . . . . 11 (𝑥 ↦ ({𝑥} 𝑁)) = (𝑧 ↦ ({𝑧} 𝑁))
48 simp-4r 783 . . . . . . . . . . . . . 14 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ∈ (SubGrp‘𝐺))
4948ad2antrr 725 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → ∈ (SubGrp‘𝐺))
50 simp-4r 783 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑥)
51 simplr 768 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑦)
52 eqid 2758 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
5352subgcl 18356 . . . . . . . . . . . . 13 (( ∈ (SubGrp‘𝐺) ∧ 𝑥𝑦) → (𝑥(+g𝐺)𝑦) ∈ )
5449, 50, 51, 53syl3anc 1368 . . . . . . . . . . . 12 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑥(+g𝐺)𝑦) ∈ )
55 sneq 4532 . . . . . . . . . . . . . . 15 (𝑧 = (𝑥(+g𝐺)𝑦) → {𝑧} = {(𝑥(+g𝐺)𝑦)})
5655oveq1d 7165 . . . . . . . . . . . . . 14 (𝑧 = (𝑥(+g𝐺)𝑦) → ({𝑧} 𝑁) = ({(𝑥(+g𝐺)𝑦)} 𝑁))
5756eqeq2d 2769 . . . . . . . . . . . . 13 (𝑧 = (𝑥(+g𝐺)𝑦) → ((𝑖(+g𝑄)𝑗) = ({𝑧} 𝑁) ↔ (𝑖(+g𝑄)𝑗) = ({(𝑥(+g𝐺)𝑦)} 𝑁)))
5857adantl 485 . . . . . . . . . . . 12 ((((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) ∧ 𝑧 = (𝑥(+g𝐺)𝑦)) → ((𝑖(+g𝑄)𝑗) = ({𝑧} 𝑁) ↔ (𝑖(+g𝑄)𝑗) = ({(𝑥(+g𝐺)𝑦)} 𝑁)))
59 simpr 488 . . . . . . . . . . . . . . . 16 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖 = ({𝑥} 𝑁))
6017adantr 484 . . . . . . . . . . . . . . . 16 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
6159, 60eqtr4d 2796 . . . . . . . . . . . . . . 15 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
6261ad2antrr 725 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
63 simpr 488 . . . . . . . . . . . . . . 15 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑗 = ({𝑦} 𝑁))
641ad4antr 731 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
6564ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
6665, 14syl 17 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
6749, 7syl 17 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝐵)
6867, 51sseldd 3893 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑦𝐵)
696, 13, 66, 68quslsm 31114 . . . . . . . . . . . . . . 15 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → [𝑦](𝐺 ~QG 𝑁) = ({𝑦} 𝑁))
7063, 69eqtr4d 2796 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑗 = [𝑦](𝐺 ~QG 𝑁))
7162, 70oveq12d 7168 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) = ([𝑥](𝐺 ~QG 𝑁)(+g𝑄)[𝑦](𝐺 ~QG 𝑁)))
729adantr 484 . . . . . . . . . . . . . . 15 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑥𝐵)
7372ad2antrr 725 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑥𝐵)
74 eqid 2758 . . . . . . . . . . . . . . 15 (+g𝑄) = (+g𝑄)
752, 6, 52, 74qusadd 18404 . . . . . . . . . . . . . 14 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝐵𝑦𝐵) → ([𝑥](𝐺 ~QG 𝑁)(+g𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝑁))
7665, 73, 68, 75syl3anc 1368 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → ([𝑥](𝐺 ~QG 𝑁)(+g𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝑁))
7767, 54sseldd 3893 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
786, 13, 66, 77quslsm 31114 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝑁) = ({(𝑥(+g𝐺)𝑦)} 𝑁))
7971, 76, 783eqtrd 2797 . . . . . . . . . . . 12 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) = ({(𝑥(+g𝐺)𝑦)} 𝑁))
8054, 58, 79rspcedvd 3544 . . . . . . . . . . 11 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → ∃𝑧 (𝑖(+g𝑄)𝑗) = ({𝑧} 𝑁))
81 ovexd 7185 . . . . . . . . . . 11 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) ∈ V)
8247, 80, 81elrnmptd 5802 . . . . . . . . . 10 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
8382adantllr 718 . . . . . . . . 9 ((((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
84 sneq 4532 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → {𝑥} = {𝑦})
8584oveq1d 7165 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} 𝑁) = ({𝑦} 𝑁))
8685cbvmptv 5135 . . . . . . . . . . . 12 (𝑥 ↦ ({𝑥} 𝑁)) = (𝑦 ↦ ({𝑦} 𝑁))
87 ovex 7183 . . . . . . . . . . . 12 ({𝑦} 𝑁) ∈ V
8886, 87elrnmpti 5801 . . . . . . . . . . 11 (𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ ∃𝑦 𝑗 = ({𝑦} 𝑁))
8988biimpi 219 . . . . . . . . . 10 (𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) → ∃𝑦 𝑗 = ({𝑦} 𝑁))
9089adantl 485 . . . . . . . . 9 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → ∃𝑦 𝑗 = ({𝑦} 𝑁))
9183, 90r19.29a 3213 . . . . . . . 8 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → (𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
9291ralrimiva 3113 . . . . . . 7 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
93 eqid 2758 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
9493subginvcl 18355 . . . . . . . . . 10 (( ∈ (SubGrp‘𝐺) ∧ 𝑥) → ((invg𝐺)‘𝑥) ∈ )
9594ad5ant24 760 . . . . . . . . 9 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝐺)‘𝑥) ∈ )
96 simpr 488 . . . . . . . . . . . . 13 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → 𝑦 = ((invg𝐺)‘𝑥))
9796sneqd 4534 . . . . . . . . . . . 12 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → {𝑦} = {((invg𝐺)‘𝑥)})
9897oveq1d 7165 . . . . . . . . . . 11 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → ({𝑦} 𝑁) = ({((invg𝐺)‘𝑥)} 𝑁))
998adantr 484 . . . . . . . . . . . . . 14 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → 𝐵)
10094ad4ant24 753 . . . . . . . . . . . . . 14 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ((invg𝐺)‘𝑥) ∈ )
10199, 100sseldd 3893 . . . . . . . . . . . . 13 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ((invg𝐺)‘𝑥) ∈ 𝐵)
1026, 13, 16, 101quslsm 31114 . . . . . . . . . . . 12 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg𝐺)‘𝑥)} 𝑁))
103102ad2antrr 725 . . . . . . . . . . 11 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg𝐺)‘𝑥)} 𝑁))
10498, 103eqtr4d 2796 . . . . . . . . . 10 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → ({𝑦} 𝑁) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
105104eqeq2d 2769 . . . . . . . . 9 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → (((invg𝑄)‘𝑖) = ({𝑦} 𝑁) ↔ ((invg𝑄)‘𝑖) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁)))
10661fveq2d 6662 . . . . . . . . . 10 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) = ((invg𝑄)‘[𝑥](𝐺 ~QG 𝑁)))
107 eqid 2758 . . . . . . . . . . . 12 (invg𝑄) = (invg𝑄)
1082, 6, 93, 107qusinv 18406 . . . . . . . . . . 11 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝐵) → ((invg𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
10964, 72, 108syl2anc 587 . . . . . . . . . 10 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
110106, 109eqtrd 2793 . . . . . . . . 9 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
11195, 105, 110rspcedvd 3544 . . . . . . . 8 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ∃𝑦 ((invg𝑄)‘𝑖) = ({𝑦} 𝑁))
112 fvexd 6673 . . . . . . . 8 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) ∈ V)
11386, 111, 112elrnmptd 5802 . . . . . . 7 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
11492, 113jca 515 . . . . . 6 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → (∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
115114adantllr 718 . . . . 5 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → (∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
116 ovex 7183 . . . . . . . 8 ({𝑥} 𝑁) ∈ V
11727, 116elrnmpti 5801 . . . . . . 7 (𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 𝑖 = ({𝑥} 𝑁))
118117biimpi 219 . . . . . 6 (𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) → ∃𝑥 𝑖 = ({𝑥} 𝑁))
119118adantl 485 . . . . 5 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → ∃𝑥 𝑖 = ({𝑥} 𝑁))
12040, 44, 115, 119r19.29af2 3253 . . . 4 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → (∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
121120ralrimiva 3113 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ∀𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
122 eqid 2758 . . . . 5 (Base‘𝑄) = (Base‘𝑄)
123122, 74, 107issubg2 18361 . . . 4 (𝑄 ∈ Grp → (ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄) ↔ (ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ↦ ({𝑥} 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))))
124123biimpar 481 . . 3 ((𝑄 ∈ Grp ∧ (ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ↦ ({𝑥} 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
1255, 29, 36, 121, 124syl13anc 1369 . 2 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
126 nsgqusf1o.t . 2 𝑇 = (SubGrp‘𝑄)
127125, 126eleqtrrdi 2863 1 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  {csn 4522   ↦ cmpt 5112  ran crn 5525  ‘cfv 6335  (class class class)co 7150  [cec 8297   / cqs 8298  Basecbs 16541  +gcplusg 16623  lecple 16630  0gc0g 16771   /s cqus 16836  toInccipo 17827  Grpcgrp 18169  invgcminusg 18170  SubGrpcsubg 18340  NrmSGrpcnsg 18341   ~QG cqg 18342  LSSumclsm 18826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-tpos 7902  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-ec 8301  df-qs 8305  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-z 12021  df-dec 12138  df-uz 12283  df-fz 12940  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-mulr 16637  df-sca 16639  df-vsca 16640  df-ip 16641  df-tset 16642  df-ple 16643  df-ds 16645  df-0g 16773  df-imas 16839  df-qus 16840  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-grp 18172  df-minusg 18173  df-subg 18343  df-nsg 18344  df-eqg 18345  df-oppg 18541  df-lsm 18828 This theorem is referenced by:  nsgqusf1olem2  31120  nsgqusf1olem3  31121
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