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Mirrors > Home > MPE Home > Th. List > itg1addlem3 | Structured version Visualization version GIF version |
Description: Lemma for itg1add 25082. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
i1fadd.1 | ⊢ (𝜑 → 𝐹 ∈ dom ∫1) |
i1fadd.2 | ⊢ (𝜑 → 𝐺 ∈ dom ∫1) |
itg1add.3 | ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) |
Ref | Expression |
---|---|
itg1addlem3 | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2741 | . . . . 5 ⊢ (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0)) | |
2 | eqeq1 2741 | . . . . 5 ⊢ (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0)) | |
3 | 1, 2 | bi2anan9 638 | . . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
4 | sneq 4601 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → {𝑖} = {𝐴}) | |
5 | 4 | imaeq2d 6018 | . . . . . 6 ⊢ (𝑖 = 𝐴 → (◡𝐹 “ {𝑖}) = (◡𝐹 “ {𝐴})) |
6 | sneq 4601 | . . . . . . 7 ⊢ (𝑗 = 𝐵 → {𝑗} = {𝐵}) | |
7 | 6 | imaeq2d 6018 | . . . . . 6 ⊢ (𝑗 = 𝐵 → (◡𝐺 “ {𝑗}) = (◡𝐺 “ {𝐵})) |
8 | 5, 7 | ineqan12d 4179 | . . . . 5 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) = ((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))) |
9 | 8 | fveq2d 6851 | . . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) |
10 | 3, 9 | ifbieq2d 4517 | . . 3 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))) |
11 | itg1add.3 | . . 3 ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))))) | |
12 | c0ex 11156 | . . . 4 ⊢ 0 ∈ V | |
13 | fvex 6860 | . . . 4 ⊢ (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))) ∈ V | |
14 | 12, 13 | ifex 4541 | . . 3 ⊢ if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) ∈ V |
15 | 10, 11, 14 | ovmpoa 7515 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))) |
16 | iffalse 4500 | . 2 ⊢ (¬ (𝐴 = 0 ∧ 𝐵 = 0) → if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) | |
17 | 15, 16 | sylan9eq 2797 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3914 ifcif 4491 {csn 4591 ◡ccnv 5637 dom cdm 5638 “ cima 5641 ‘cfv 6501 (class class class)co 7362 ∈ cmpo 7364 ℝcr 11057 0cc0 11058 volcvol 24843 ∫1citg1 24995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-mulcl 11120 ax-i2m1 11126 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 |
This theorem is referenced by: itg1addlem4 25079 itg1addlem4OLD 25080 itg1addlem5 25081 |
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