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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sxval | Structured version Visualization version GIF version | ||
| Description: Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.) | 
| Ref | Expression | 
|---|---|
| sxval.1 | ⊢ 𝐴 = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) | 
| Ref | Expression | 
|---|---|
| sxval | ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elex 3501 | . . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
| 2 | elex 3501 | . . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V) | |
| 3 | id 22 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆) | |
| 4 | eqidd 2738 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑡 = 𝑡) | |
| 5 | eqidd 2738 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦)) | |
| 6 | 3, 4, 5 | mpoeq123dv 7508 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) | 
| 7 | 6 | rneqd 5949 | . . . . 5 ⊢ (𝑠 = 𝑆 → ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) | 
| 8 | 7 | fveq2d 6910 | . . . 4 ⊢ (𝑠 = 𝑆 → (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))) | 
| 9 | eqidd 2738 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑆 = 𝑆) | |
| 10 | id 22 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇) | |
| 11 | eqidd 2738 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦)) | |
| 12 | 9, 10, 11 | mpoeq123dv 7508 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) | 
| 13 | 12 | rneqd 5949 | . . . . 5 ⊢ (𝑡 = 𝑇 → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) | 
| 14 | 13 | fveq2d 6910 | . . . 4 ⊢ (𝑡 = 𝑇 → (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))) | 
| 15 | df-sx 34190 | . . . 4 ⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))) | |
| 16 | fvex 6919 | . . . 4 ⊢ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) ∈ V | |
| 17 | 8, 14, 15, 16 | ovmpo 7593 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))) | 
| 18 | 1, 2, 17 | syl2an 596 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))) | 
| 19 | sxval.1 | . . 3 ⊢ 𝐴 = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) | |
| 20 | 19 | fveq2i 6909 | . 2 ⊢ (sigaGen‘𝐴) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) | 
| 21 | 18, 20 | eqtr4di 2795 | 1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 × cxp 5683 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 sigaGencsigagen 34139 ×s csx 34189 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sx 34190 | 
| This theorem is referenced by: sxsiga 34192 sxsigon 34193 elsx 34195 mbfmco2 34267 sxbrsigalem5 34290 sxbrsiga 34292 | 
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