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Mirrors > Home > MPE Home > Th. List > itg1addlem3 | Structured version Visualization version GIF version |
Description: Lemma for itg1add 23809. (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 2803 | . . . . 5 ⊢ (𝑖 = 𝐴 → (𝑖 = 0 ↔ 𝐴 = 0)) | |
2 | eqeq1 2803 | . . . . 5 ⊢ (𝑗 = 𝐵 → (𝑗 = 0 ↔ 𝐵 = 0)) | |
3 | 1, 2 | bi2anan9 630 | . . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((𝑖 = 0 ∧ 𝑗 = 0) ↔ (𝐴 = 0 ∧ 𝐵 = 0))) |
4 | sneq 4378 | . . . . . . 7 ⊢ (𝑖 = 𝐴 → {𝑖} = {𝐴}) | |
5 | 4 | imaeq2d 5683 | . . . . . 6 ⊢ (𝑖 = 𝐴 → (◡𝐹 “ {𝑖}) = (◡𝐹 “ {𝐴})) |
6 | sneq 4378 | . . . . . . 7 ⊢ (𝑗 = 𝐵 → {𝑗} = {𝐵}) | |
7 | 6 | imaeq2d 5683 | . . . . . 6 ⊢ (𝑗 = 𝐵 → (◡𝐺 “ {𝑗}) = (◡𝐺 “ {𝐵})) |
8 | 5, 7 | ineqan12d 4014 | . . . . 5 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → ((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})) = ((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))) |
9 | 8 | fveq2d 6415 | . . . 4 ⊢ ((𝑖 = 𝐴 ∧ 𝑗 = 𝐵) → (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗}))) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) |
10 | 3, 9 | ifbieq2d 4302 | . . 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 10322 | . . . 4 ⊢ 0 ∈ V | |
13 | fvex 6424 | . . . 4 ⊢ (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))) ∈ V | |
14 | 12, 13 | ifex 4325 | . . 3 ⊢ if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵})))) ∈ V |
15 | 10, 11, 14 | ovmpt2a 7025 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐼𝐵) = if((𝐴 = 0 ∧ 𝐵 = 0), 0, (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))) |
16 | iffalse 4286 | . 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 2853 | 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 385 = wceq 1653 ∈ wcel 2157 ∩ cin 3768 ifcif 4277 {csn 4368 ◡ccnv 5311 dom cdm 5312 “ cima 5315 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 ℝcr 10223 0cc0 10224 volcvol 23571 ∫1citg1 23723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-mulcl 10286 ax-i2m1 10292 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 |
This theorem is referenced by: itg1addlem4 23807 itg1addlem5 23808 |
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