MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasrng Structured version   Visualization version   GIF version

Theorem imasrng 20195
Description: The image structure of a non-unital ring is a non-unital ring (imasring 20344 analog). (Contributed by AV, 22-Feb-2025.)
Hypotheses
Ref Expression
imasrng.u (𝜑𝑈 = (𝐹s 𝑅))
imasrng.v (𝜑𝑉 = (Base‘𝑅))
imasrng.p + = (+g𝑅)
imasrng.t · = (.r𝑅)
imasrng.f (𝜑𝐹:𝑉onto𝐵)
imasrng.e1 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasrng.e2 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
imasrng.r (𝜑𝑅 ∈ Rng)
Assertion
Ref Expression
imasrng (𝜑𝑈 ∈ Rng)
Distinct variable groups:   + ,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞   𝑈,𝑎,𝑏,𝑝,𝑞   𝐵,𝑎,𝑏,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞   𝑅,𝑎,𝑏,𝑝,𝑞   𝑉,𝑎,𝑏,𝑝,𝑞   · ,𝑝,𝑞
Allowed substitution hints:   + (𝑎,𝑏)   · (𝑎,𝑏)

Proof of Theorem imasrng
Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasrng.u . . 3 (𝜑𝑈 = (𝐹s 𝑅))
2 imasrng.v . . 3 (𝜑𝑉 = (Base‘𝑅))
3 imasrng.f . . 3 (𝜑𝐹:𝑉onto𝐵)
4 imasrng.r . . 3 (𝜑𝑅 ∈ Rng)
51, 2, 3, 4imasbas 17559 . 2 (𝜑𝐵 = (Base‘𝑈))
6 eqidd 2736 . 2 (𝜑 → (+g𝑈) = (+g𝑈))
7 eqidd 2736 . 2 (𝜑 → (.r𝑈) = (.r𝑈))
8 imasrng.p . . . . 5 + = (+g𝑅)
98a1i 11 . . . 4 (𝜑+ = (+g𝑅))
10 imasrng.e1 . . . 4 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
11 rngabl 20173 . . . . 5 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
124, 11syl 17 . . . 4 (𝜑𝑅 ∈ Abel)
13 eqid 2735 . . . 4 (0g𝑅) = (0g𝑅)
141, 2, 9, 3, 10, 12, 13imasabl 19909 . . 3 (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘(0g𝑅)) = (0g𝑈)))
1514simpld 494 . 2 (𝜑𝑈 ∈ Abel)
16 imasrng.e2 . . . 4 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))
17 imasrng.t . . . 4 · = (.r𝑅)
18 eqid 2735 . . . 4 (.r𝑈) = (.r𝑈)
194adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑅 ∈ Rng)
20 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢𝑉)
212adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑉 = (Base‘𝑅))
2220, 21eleqtrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑢 ∈ (Base‘𝑅))
23 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣𝑉)
2423, 21eleqtrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → 𝑣 ∈ (Base‘𝑅))
25 eqid 2735 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
2625, 17rngcl 20182 . . . . . . 7 ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2719, 22, 24, 26syl3anc 1370 . . . . . 6 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ (Base‘𝑅))
2827, 21eleqtrrd 2842 . . . . 5 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 · 𝑣) ∈ 𝑉)
2928caovclg 7625 . . . 4 ((𝜑 ∧ (𝑝𝑉𝑞𝑉)) → (𝑝 · 𝑞) ∈ 𝑉)
303, 16, 1, 2, 4, 17, 18, 29imasmulf 17583 . . 3 (𝜑 → (.r𝑈):(𝐵 × 𝐵)⟶𝐵)
3130fovcld 7560 . 2 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢(.r𝑈)𝑣) ∈ 𝐵)
32 forn 6824 . . . . . . . . 9 (𝐹:𝑉onto𝐵 → ran 𝐹 = 𝐵)
333, 32syl 17 . . . . . . . 8 (𝜑 → ran 𝐹 = 𝐵)
3433eleq2d 2825 . . . . . . 7 (𝜑 → (𝑢 ∈ ran 𝐹𝑢𝐵))
3533eleq2d 2825 . . . . . . 7 (𝜑 → (𝑣 ∈ ran 𝐹𝑣𝐵))
3633eleq2d 2825 . . . . . . 7 (𝜑 → (𝑤 ∈ ran 𝐹𝑤𝐵))
3734, 35, 363anbi123d 1435 . . . . . 6 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (𝑢𝐵𝑣𝐵𝑤𝐵)))
38 fofn 6823 . . . . . . 7 (𝐹:𝑉onto𝐵𝐹 Fn 𝑉)
39 fvelrnb 6969 . . . . . . . 8 (𝐹 Fn 𝑉 → (𝑢 ∈ ran 𝐹 ↔ ∃𝑥𝑉 (𝐹𝑥) = 𝑢))
40 fvelrnb 6969 . . . . . . . 8 (𝐹 Fn 𝑉 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑦𝑉 (𝐹𝑦) = 𝑣))
41 fvelrnb 6969 . . . . . . . 8 (𝐹 Fn 𝑉 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4239, 40, 413anbi123d 1435 . . . . . . 7 (𝐹 Fn 𝑉 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
433, 38, 423syl 18 . . . . . 6 (𝜑 → ((𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐹𝑤 ∈ ran 𝐹) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
4437, 43bitr3d 281 . . . . 5 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤)))
45 3reeanv 3228 . . . . 5 (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) ↔ (∃𝑥𝑉 (𝐹𝑥) = 𝑢 ∧ ∃𝑦𝑉 (𝐹𝑦) = 𝑣 ∧ ∃𝑧𝑉 (𝐹𝑧) = 𝑤))
4644, 45bitr4di 289 . . . 4 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) ↔ ∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤)))
474adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑅 ∈ Rng)
48 simp2 1136 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥𝑉)
4923ad2ant1 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑉 = (Base‘𝑅))
5048, 49eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑉𝑦𝑉) → 𝑥 ∈ (Base‘𝑅))
51503adant3r3 1183 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥 ∈ (Base‘𝑅))
52 simp3 1137 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦𝑉)
5352, 49eleqtrd 2841 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑉𝑦𝑉) → 𝑦 ∈ (Base‘𝑅))
54533adant3r3 1183 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑦 ∈ (Base‘𝑅))
55 simpr3 1195 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧𝑉)
562adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑉 = (Base‘𝑅))
5755, 56eleqtrd 2841 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑧 ∈ (Base‘𝑅))
5825, 17rngass 20177 . . . . . . . . . . . . 13 ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
5947, 51, 54, 57, 58syl13anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))
6059fveq2d 6911 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 · 𝑦) · 𝑧)) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
61 simpl 482 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝜑)
6228caovclg 7625 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
63623adantr3 1170 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑦) ∈ 𝑉)
643, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
6561, 63, 55, 64syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 · 𝑦) · 𝑧)))
66 simpr1 1193 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑥𝑉)
6728caovclg 7625 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
68673adantr1 1168 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 · 𝑧) ∈ 𝑉)
693, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . 12 ((𝜑𝑥𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7061, 66, 68, 69syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘(𝑥 · (𝑦 · 𝑧))))
7160, 65, 703eqtr4d 2785 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
723, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . 12 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
73723adant3r3 1183 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 · 𝑦)))
7473oveq1d 7446 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑦))(.r𝑈)(𝐹𝑧)))
753, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . 12 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
76753adant3r1 1181 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 · 𝑧)))
7776oveq2d 7447 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 · 𝑧))))
7871, 74, 773eqtr4d 2785 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
79 simp1 1135 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑥) = 𝑢)
80 simp2 1136 . . . . . . . . . . . 12 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑦) = 𝑣)
8179, 80oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑦)) = (𝑢(.r𝑈)𝑣))
82 simp3 1137 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝐹𝑧) = 𝑤)
8381, 82oveq12d 7449 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤))
8480, 82oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(.r𝑈)(𝐹𝑧)) = (𝑣(.r𝑈)𝑤))
8579, 84oveq12d 7449 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))
8683, 85eqeq12d 2751 . . . . . . . . 9 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((((𝐹𝑥)(.r𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) ↔ ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
8778, 86syl5ibcom 245 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
88873exp2 1353 . . . . . . 7 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))))))
8988imp32 418 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))))
9089rexlimdv 3151 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9190rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9246, 91sylbid 240 . . 3 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤))))
9392imp 406 . 2 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(.r𝑈)𝑣)(.r𝑈)𝑤) = (𝑢(.r𝑈)(𝑣(.r𝑈)𝑤)))
9425, 8, 17rngdi 20178 . . . . . . . . . . . . 13 ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9547, 51, 54, 57, 94syl13anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))
9695fveq2d 6911 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘(𝑥 · (𝑦 + 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
9725, 8rngacl 20180 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Rng ∧ 𝑢 ∈ (Base‘𝑅) ∧ 𝑣 ∈ (Base‘𝑅)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
9819, 22, 24, 97syl3anc 1370 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ (Base‘𝑅))
9998, 21eleqtrrd 2842 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑢𝑉𝑣𝑉)) → (𝑢 + 𝑣) ∈ 𝑉)
10099caovclg 7625 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1011003adantr1 1168 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑦 + 𝑧) ∈ 𝑉)
1023, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . 12 ((𝜑𝑥𝑉 ∧ (𝑦 + 𝑧) ∈ 𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10361, 66, 101, 102syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = (𝐹‘(𝑥 · (𝑦 + 𝑧))))
10428caovclg 7625 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
1051043adantr2 1169 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 · 𝑧) ∈ 𝑉)
106 eqid 2735 . . . . . . . . . . . . 13 (+g𝑈) = (+g𝑈)
1073, 10, 1, 2, 4, 8, 106imasaddval 17579 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 · 𝑦) ∈ 𝑉 ∧ (𝑥 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
10861, 63, 105, 107syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))) = (𝐹‘((𝑥 · 𝑦) + (𝑥 · 𝑧))))
10996, 103, 1083eqtr4d 2785 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
1103, 10, 1, 2, 4, 8, 106imasaddval 17579 . . . . . . . . . . . 12 ((𝜑𝑦𝑉𝑧𝑉) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
1111103adant3r1 1181 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝐹‘(𝑦 + 𝑧)))
112111oveq2d 7447 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = ((𝐹𝑥)(.r𝑈)(𝐹‘(𝑦 + 𝑧))))
1133, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . 12 ((𝜑𝑥𝑉𝑧𝑉) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
1141133adant3r2 1182 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝐹‘(𝑥 · 𝑧)))
11573, 114oveq12d 7449 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑦))(+g𝑈)(𝐹‘(𝑥 · 𝑧))))
116109, 112, 1153eqtr4d 2785 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))))
11780, 82oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑦)(+g𝑈)(𝐹𝑧)) = (𝑣(+g𝑈)𝑤))
11879, 117oveq12d 7449 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)))
11979, 82oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(.r𝑈)(𝐹𝑧)) = (𝑢(.r𝑈)𝑤))
12081, 119oveq12d 7449 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))
121118, 120eqeq12d 2751 . . . . . . . . 9 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)((𝐹𝑦)(+g𝑈)(𝐹𝑧))) = (((𝐹𝑥)(.r𝑈)(𝐹𝑦))(+g𝑈)((𝐹𝑥)(.r𝑈)(𝐹𝑧))) ↔ (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
122116, 121syl5ibcom 245 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
1231223exp2 1353 . . . . . . 7 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))))))
124123imp32 418 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))))
125124rexlimdv 3151 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
126125rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
12746, 126sylbid 240 . . 3 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤))))
128127imp 406 . 2 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢(.r𝑈)(𝑣(+g𝑈)𝑤)) = ((𝑢(.r𝑈)𝑣)(+g𝑈)(𝑢(.r𝑈)𝑤)))
12925, 8, 17rngdir 20179 . . . . . . . . . . . . 13 ((𝑅 ∈ Rng ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
13047, 51, 54, 57, 129syl13anc 1371 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
131130fveq2d 6911 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝐹‘((𝑥 + 𝑦) · 𝑧)) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
13299caovclg 7625 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1331323adantr3 1170 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
1343, 16, 1, 2, 4, 17, 18imasmulval 17582 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 + 𝑦) ∈ 𝑉𝑧𝑉) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
13561, 133, 55, 134syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = (𝐹‘((𝑥 + 𝑦) · 𝑧)))
1363, 10, 1, 2, 4, 8, 106imasaddval 17579 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 · 𝑧) ∈ 𝑉 ∧ (𝑦 · 𝑧) ∈ 𝑉) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
13761, 105, 68, 136syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))) = (𝐹‘((𝑥 · 𝑧) + (𝑦 · 𝑧))))
138131, 135, 1373eqtr4d 2785 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
1393, 10, 1, 2, 4, 8, 106imasaddval 17579 . . . . . . . . . . . 12 ((𝜑𝑥𝑉𝑦𝑉) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
1401393adant3r3 1183 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝐹‘(𝑥 + 𝑦)))
141140oveq1d 7446 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝐹‘(𝑥 + 𝑦))(.r𝑈)(𝐹𝑧)))
142114, 76oveq12d 7449 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝐹‘(𝑥 · 𝑧))(+g𝑈)(𝐹‘(𝑦 · 𝑧))))
143138, 141, 1423eqtr4d 2785 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))))
14479, 80oveq12d 7449 . . . . . . . . . . 11 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) = (𝑢(+g𝑈)𝑣))
145144, 82oveq12d 7449 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤))
146119, 84oveq12d 7449 . . . . . . . . . 10 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))
147145, 146eqeq12d 2751 . . . . . . . . 9 (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((((𝐹𝑥)(+g𝑈)(𝐹𝑦))(.r𝑈)(𝐹𝑧)) = (((𝐹𝑥)(.r𝑈)(𝐹𝑧))(+g𝑈)((𝐹𝑦)(.r𝑈)(𝐹𝑧))) ↔ ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
148143, 147syl5ibcom 245 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
1491483exp2 1353 . . . . . . 7 (𝜑 → (𝑥𝑉 → (𝑦𝑉 → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))))))
150149imp32 418 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑧𝑉 → (((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))))
151150rexlimdv 3151 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (∃𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
152151rexlimdvva 3211 . . . 4 (𝜑 → (∃𝑥𝑉𝑦𝑉𝑧𝑉 ((𝐹𝑥) = 𝑢 ∧ (𝐹𝑦) = 𝑣 ∧ (𝐹𝑧) = 𝑤) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
15346, 152sylbid 240 . . 3 (𝜑 → ((𝑢𝐵𝑣𝐵𝑤𝐵) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤))))
154153imp 406 . 2 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑈)𝑣)(.r𝑈)𝑤) = ((𝑢(.r𝑈)𝑤)(+g𝑈)(𝑣(.r𝑈)𝑤)))
1555, 6, 7, 15, 31, 93, 128, 154isrngd 20191 1 (𝜑𝑈 ∈ Rng)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  ran crn 5690   Fn wfn 6558  ontowfo 6561  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  0gc0g 17486  s cimas 17551  Abelcabl 19814  Rngcrng 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171
This theorem is referenced by:  imasrngf1  20196  qusrng  20198
  Copyright terms: Public domain W3C validator