Proof of Theorem erov
Step | Hyp | Ref
| Expression |
1 | | eropr.1 |
. . . . 5
⊢ 𝐽 = (𝐴 / 𝑅) |
2 | | eropr.2 |
. . . . 5
⊢ 𝐾 = (𝐵 / 𝑆) |
3 | | eropr.3 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ 𝑍) |
4 | | eropr.4 |
. . . . 5
⊢ (𝜑 → 𝑅 Er 𝑈) |
5 | | eropr.5 |
. . . . 5
⊢ (𝜑 → 𝑆 Er 𝑉) |
6 | | eropr.6 |
. . . . 5
⊢ (𝜑 → 𝑇 Er 𝑊) |
7 | | eropr.7 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
8 | | eropr.8 |
. . . . 5
⊢ (𝜑 → 𝐵 ⊆ 𝑉) |
9 | | eropr.9 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ 𝑊) |
10 | | eropr.10 |
. . . . 5
⊢ (𝜑 → + :(𝐴 × 𝐵)⟶𝐶) |
11 | | eropr.11 |
. . . . 5
⊢ ((𝜑 ∧ ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑡 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵))) → ((𝑟𝑅𝑠 ∧ 𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢))) |
12 | | eropr.12 |
. . . . 5
⊢ ⨣ =
{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12 | erovlem 8495 |
. . . 4
⊢ (𝜑 → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))) |
14 | 13 | 3ad2ant1 1135 |
. . 3
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ⨣ = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))) |
15 | | simprl 771 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → 𝑥 = [𝑃]𝑅) |
16 | 15 | eqeq1d 2739 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (𝑥 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑝]𝑅)) |
17 | | simprr 773 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → 𝑦 = [𝑄]𝑆) |
18 | 17 | eqeq1d 2739 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (𝑦 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑞]𝑆)) |
19 | 16, 18 | anbi12d 634 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆))) |
20 | 19 | anbi1d 633 |
. . . . 5
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) |
21 | 20 | 2rexbidv 3219 |
. . . 4
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) |
22 | 21 | iotabidv 6364 |
. . 3
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ (𝑥 = [𝑃]𝑅 ∧ 𝑦 = [𝑄]𝑆)) → (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 ((𝑥 = [𝑝]𝑅 ∧ 𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) |
23 | | eropr.13 |
. . . . 5
⊢ (𝜑 → 𝑅 ∈ 𝑋) |
24 | | ecelqsg 8454 |
. . . . . 6
⊢ ((𝑅 ∈ 𝑋 ∧ 𝑃 ∈ 𝐴) → [𝑃]𝑅 ∈ (𝐴 / 𝑅)) |
25 | 24, 1 | eleqtrrdi 2849 |
. . . . 5
⊢ ((𝑅 ∈ 𝑋 ∧ 𝑃 ∈ 𝐴) → [𝑃]𝑅 ∈ 𝐽) |
26 | 23, 25 | sylan 583 |
. . . 4
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴) → [𝑃]𝑅 ∈ 𝐽) |
27 | 26 | 3adant3 1134 |
. . 3
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → [𝑃]𝑅 ∈ 𝐽) |
28 | | eropr.14 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ 𝑌) |
29 | | ecelqsg 8454 |
. . . . . 6
⊢ ((𝑆 ∈ 𝑌 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ (𝐵 / 𝑆)) |
30 | 29, 2 | eleqtrrdi 2849 |
. . . . 5
⊢ ((𝑆 ∈ 𝑌 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ 𝐾) |
31 | 28, 30 | sylan 583 |
. . . 4
⊢ ((𝜑 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ 𝐾) |
32 | 31 | 3adant2 1133 |
. . 3
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → [𝑄]𝑆 ∈ 𝐾) |
33 | | iotaex 6360 |
. . . 4
⊢
(℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V |
34 | 33 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V) |
35 | 14, 22, 27, 32, 34 | ovmpod 7361 |
. 2
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ([𝑃]𝑅 ⨣ [𝑄]𝑆) = (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) |
36 | | eqid 2737 |
. . . . . . 7
⊢ [𝑃]𝑅 = [𝑃]𝑅 |
37 | | eqid 2737 |
. . . . . . 7
⊢ [𝑄]𝑆 = [𝑄]𝑆 |
38 | 36, 37 | pm3.2i 474 |
. . . . . 6
⊢ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) |
39 | | eqid 2737 |
. . . . . 6
⊢ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇 |
40 | 38, 39 | pm3.2i 474 |
. . . . 5
⊢ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇) |
41 | | eceq1 8429 |
. . . . . . . . 9
⊢ (𝑝 = 𝑃 → [𝑝]𝑅 = [𝑃]𝑅) |
42 | 41 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑝 = 𝑃 → ([𝑃]𝑅 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑃]𝑅)) |
43 | 42 | anbi1d 633 |
. . . . . . 7
⊢ (𝑝 = 𝑃 → (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆))) |
44 | | oveq1 7220 |
. . . . . . . . 9
⊢ (𝑝 = 𝑃 → (𝑝 + 𝑞) = (𝑃 + 𝑞)) |
45 | 44 | eceq1d 8430 |
. . . . . . . 8
⊢ (𝑝 = 𝑃 → [(𝑝 + 𝑞)]𝑇 = [(𝑃 + 𝑞)]𝑇) |
46 | 45 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑝 = 𝑃 → ([(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇)) |
47 | 43, 46 | anbi12d 634 |
. . . . . 6
⊢ (𝑝 = 𝑃 → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇))) |
48 | | eceq1 8429 |
. . . . . . . . 9
⊢ (𝑞 = 𝑄 → [𝑞]𝑆 = [𝑄]𝑆) |
49 | 48 | eqeq2d 2748 |
. . . . . . . 8
⊢ (𝑞 = 𝑄 → ([𝑄]𝑆 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑄]𝑆)) |
50 | 49 | anbi2d 632 |
. . . . . . 7
⊢ (𝑞 = 𝑄 → (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆))) |
51 | | oveq2 7221 |
. . . . . . . . 9
⊢ (𝑞 = 𝑄 → (𝑃 + 𝑞) = (𝑃 + 𝑄)) |
52 | 51 | eceq1d 8430 |
. . . . . . . 8
⊢ (𝑞 = 𝑄 → [(𝑃 + 𝑞)]𝑇 = [(𝑃 + 𝑄)]𝑇) |
53 | 52 | eqeq2d 2748 |
. . . . . . 7
⊢ (𝑞 = 𝑄 → ([(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)) |
54 | 50, 53 | anbi12d 634 |
. . . . . 6
⊢ (𝑞 = 𝑄 → ((([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇))) |
55 | 47, 54 | rspc2ev 3549 |
. . . . 5
⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵 ∧ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)) |
56 | 40, 55 | mp3an3 1452 |
. . . 4
⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)) |
57 | 56 | 3adant1 1132 |
. . 3
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)) |
58 | | ecexg 8395 |
. . . . . 6
⊢ (𝑇 ∈ 𝑍 → [(𝑃 + 𝑄)]𝑇 ∈ V) |
59 | 3, 58 | syl 17 |
. . . . 5
⊢ (𝜑 → [(𝑃 + 𝑄)]𝑇 ∈ V) |
60 | 59 | 3ad2ant1 1135 |
. . . 4
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → [(𝑃 + 𝑄)]𝑇 ∈ V) |
61 | | simp1 1138 |
. . . . 5
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → 𝜑) |
62 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | eroveu 8494 |
. . . . 5
⊢ ((𝜑 ∧ ([𝑃]𝑅 ∈ 𝐽 ∧ [𝑄]𝑆 ∈ 𝐾)) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) |
63 | 61, 27, 32, 62 | syl12anc 837 |
. . . 4
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ∃!𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) |
64 | | simpr 488 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → 𝑧 = [(𝑃 + 𝑄)]𝑇) |
65 | 64 | eqeq1d 2739 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)) |
66 | 65 | anbi2d 632 |
. . . . 5
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))) |
67 | 66 | 2rexbidv 3219 |
. . . 4
⊢ (((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))) |
68 | 60, 63, 67 | iota2d 6368 |
. . 3
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇)) |
69 | 57, 68 | mpbid 235 |
. 2
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → (℩𝑧∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇) |
70 | 35, 69 | eqtrd 2777 |
1
⊢ ((𝜑 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐵) → ([𝑃]𝑅 ⨣ [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇) |