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Theorem i1fmul 24401
 Description: The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
Hypotheses
Ref Expression
Assertion
Ref Expression
i1fmul (𝜑 → (𝐹f · 𝐺) ∈ dom ∫1)

Proof of Theorem i1fmul
Dummy variables 𝑦 𝑧 𝑤 𝑣 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 remulcl 10665 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
21adantl 485 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
3 i1fadd.1 . . . 4 (𝜑𝐹 ∈ dom ∫1)
4 i1ff 24381 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 24381 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 10671 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4125 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7427 . 2 (𝜑 → (𝐹f · 𝐺):ℝ⟶ℝ)
13 i1frn 24382 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 24382 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 8827 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 587 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2758 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20 ovex 7188 . . . . . 6 (𝑢 · 𝑣) ∈ V
2119, 20fnmpoi 7777 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6586 . . . . 5 ((𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺) ↔ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
2321, 22mpbi 233 . . . 4 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
24 fofi 8848 . . . 4 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))) → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
2518, 23, 24sylancl 589 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26 eqid 2758 . . . . . . . . 9 (𝑥 · 𝑦) = (𝑥 · 𝑦)
27 rspceov 7202 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 · 𝑦) = (𝑥 · 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
2826, 27mp3an3 1447 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29 ovex 7188 . . . . . . . . 9 (𝑥 · 𝑦) ∈ V
30 eqeq1 2762 . . . . . . . . . 10 (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31302rexbidv 3224 . . . . . . . . 9 (𝑤 = (𝑥 · 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣)))
3229, 31elab 3590 . . . . . . . 8 ((𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)} ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
3328, 32sylibr 237 . . . . . . 7 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
3433adantl 485 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
355ffnd 6503 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6514 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 221 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6503 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6514 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 221 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7427 . . . . 5 (𝜑 → (𝐹f · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4241frnd 6509 . . . 4 (𝜑 → ran (𝐹f · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4319rnmpo 7284 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
4442, 43sseqtrrdi 3945 . . 3 (𝜑 → ran (𝐹f · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
4525, 44ssfid 8783 . 2 (𝜑 → ran (𝐹f · 𝐺) ∈ Fin)
4612frnd 6509 . . . . . . 7 (𝜑 → ran (𝐹f · 𝐺) ⊆ ℝ)
47 ax-resscn 10637 . . . . . . 7 ℝ ⊆ ℂ
4846, 47sstrdi 3906 . . . . . 6 (𝜑 → ran (𝐹f · 𝐺) ⊆ ℂ)
4948ssdifd 4048 . . . . 5 (𝜑 → (ran (𝐹f · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
5049sselda 3894 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
513, 6i1fmullem 24399 . . . 4 ((𝜑𝑦 ∈ (ℂ ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
5250, 51syldan 594 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
53 difss 4039 . . . . . 6 (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
54 ssfi 8747 . . . . . 6 ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
5516, 53, 54sylancl 589 . . . . 5 (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
56 i1fima 24383 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
573, 56syl 17 . . . . . . 7 (𝜑 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
58 i1fima 24383 . . . . . . . 8 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
596, 58syl 17 . . . . . . 7 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
60 inmbl 24247 . . . . . . 7 (((𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6157, 59, 60syl2anc 587 . . . . . 6 (𝜑 → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6261ralrimivw 3114 . . . . 5 (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
63 finiunmbl 24249 . . . . 5 (((ran 𝐺 ∖ {0}) ∈ Fin ∧ ∀𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol) → 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6455, 62, 63syl2anc 587 . . . 4 (𝜑 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6564adantr 484 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6652, 65eqeltrd 2852 . 2 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) ∈ dom vol)
67 mblvol 24235 . . . 4 (((𝐹f · 𝐺) “ {𝑦}) ∈ dom vol → (vol‘((𝐹f · 𝐺) “ {𝑦})) = (vol*‘((𝐹f · 𝐺) “ {𝑦})))
6866, 67syl 17 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol‘((𝐹f · 𝐺) “ {𝑦})) = (vol*‘((𝐹f · 𝐺) “ {𝑦})))
69 mblss 24236 . . . . 5 (((𝐹f · 𝐺) “ {𝑦}) ∈ dom vol → ((𝐹f · 𝐺) “ {𝑦}) ⊆ ℝ)
7066, 69syl 17 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) ⊆ ℝ)
7155adantr 484 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (ran 𝐺 ∖ {0}) ∈ Fin)
72 inss2 4136 . . . . . . 7 ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
7372a1i 11 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
7459ad2antrr 725 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
75 mblss 24236 . . . . . . 7 ((𝐺 “ {𝑧}) ∈ dom vol → (𝐺 “ {𝑧}) ⊆ ℝ)
7674, 75syl 17 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
77 mblvol 24235 . . . . . . . 8 ((𝐺 “ {𝑧}) ∈ dom vol → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
7874, 77syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
796adantr 484 . . . . . . . 8 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
80 i1fima2sn 24385 . . . . . . . 8 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8179, 80sylan 583 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8278, 81eqeltrrd 2853 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
83 ovolsscl 24191 . . . . . 6 ((((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8473, 76, 82, 83syl3anc 1368 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8571, 84fsumrecl 15144 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8652fveq2d 6666 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
87 mblss 24236 . . . . . . . . . 10 (((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
8861, 87syl 17 . . . . . . . . 9 (𝜑 → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
8988ad2antrr 725 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
9089, 84jca 515 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
9190ralrimiva 3113 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ∀𝑧 ∈ (ran 𝐺 ∖ {0})(((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
92 ovolfiniun 24206 . . . . . 6 (((ran 𝐺 ∖ {0}) ∈ Fin ∧ ∀𝑧 ∈ (ran 𝐺 ∖ {0})(((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)) → (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
9371, 91, 92syl2anc 587 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
9486, 93eqbrtrd 5057 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
95 ovollecl 24188 . . . 4 ((((𝐹f · 𝐺) “ {𝑦}) ⊆ ℝ ∧ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ ∧ (vol*‘((𝐹f · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9670, 85, 94, 95syl3anc 1368 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9768, 96eqeltrd 2852 . 2 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9812, 45, 66, 97i1fd 24386 1 (𝜑 → (𝐹f · 𝐺) ∈ dom ∫1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  {csn 4525  ∪ ciun 4886   class class class wbr 5035   × cxp 5525  ◡ccnv 5526  dom cdm 5527  ran crn 5528   “ cima 5530   Fn wfn 6334  ⟶wf 6335  –onto→wfo 6337  ‘cfv 6339  (class class class)co 7155   ∈ cmpo 7157   ∘f cof 7408  Fincfn 8532  ℂcc 10578  ℝcr 10579  0cc0 10580   · cmul 10585   ≤ cle 10719   / cdiv 11340  Σcsu 15095  vol*covol 24167  volcvol 24168  ∫1citg1 24320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-inf2 9142  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657  ax-pre-sup 10658 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7410  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-2o 8118  df-er 8304  df-map 8423  df-pm 8424  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-sup 8944  df-inf 8945  df-oi 9012  df-dju 9368  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-div 11341  df-nn 11680  df-2 11742  df-3 11743  df-n0 11940  df-z 12026  df-uz 12288  df-q 12394  df-rp 12436  df-xadd 12554  df-ioo 12788  df-ico 12790  df-icc 12791  df-fz 12945  df-fzo 13088  df-fl 13216  df-seq 13424  df-exp 13485  df-hash 13746  df-cj 14511  df-re 14512  df-im 14513  df-sqrt 14647  df-abs 14648  df-clim 14898  df-sum 15096  df-xmet 20164  df-met 20165  df-ovol 24169  df-vol 24170  df-mbf 24324  df-itg1 24325 This theorem is referenced by:  mbfmullem2  24429  ftc1anclem3  35438
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