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Theorem addsdilem4 28180
Description: Lemma for addsdi 28181. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)
Hypotheses
Ref Expression
addsdilem4.1 (𝜑𝐴 No )
addsdilem4.2 (𝜑𝐵 No )
addsdilem4.3 (𝜑𝐶 No )
addsdilem4.4 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
addsdilem4.5 (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
addsdilem4.6 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
addsdilem4.7 (𝜓𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
addsdilem4.8 (𝜓𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
Assertion
Ref Expression
addsdilem4 ((𝜑𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
Distinct variable groups:   𝐴,𝑥𝑂,𝑧𝑂   𝐵,𝑥𝑂,𝑧𝑂   𝐶,𝑥𝑂,𝑧𝑂   𝑋,𝑥𝑂,𝑧𝑂   𝑍,𝑧𝑂
Allowed substitution hints:   𝜑(𝑥𝑂,𝑧𝑂)   𝜓(𝑥𝑂,𝑧𝑂)   𝑍(𝑥𝑂)

Proof of Theorem addsdilem4
StepHypRef Expression
1 oveq1 7438 . . . . . 6 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2 oveq1 7438 . . . . . . 7 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3 oveq1 7438 . . . . . . 7 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
42, 3oveq12d 7449 . . . . . 6 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
51, 4eqeq12d 2753 . . . . 5 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) ↔ (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶))))
6 addsdilem4.4 . . . . . 6 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
76adantr 480 . . . . 5 ((𝜑𝜓) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
8 addsdilem4.7 . . . . . 6 (𝜓𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
98adantl 481 . . . . 5 ((𝜑𝜓) → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
105, 7, 9rspcdva 3623 . . . 4 ((𝜑𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11 oveq2 7439 . . . . . . 7 (𝑧𝑂 = 𝑍 → (𝐵 +s 𝑧𝑂) = (𝐵 +s 𝑍))
1211oveq2d 7447 . . . . . 6 (𝑧𝑂 = 𝑍 → (𝐴 ·s (𝐵 +s 𝑧𝑂)) = (𝐴 ·s (𝐵 +s 𝑍)))
13 oveq2 7439 . . . . . . 7 (𝑧𝑂 = 𝑍 → (𝐴 ·s 𝑧𝑂) = (𝐴 ·s 𝑍))
1413oveq2d 7447 . . . . . 6 (𝑧𝑂 = 𝑍 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
1512, 14eqeq12d 2753 . . . . 5 (𝑧𝑂 = 𝑍 → ((𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) ↔ (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
16 addsdilem4.5 . . . . . 6 (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
1716adantr 480 . . . . 5 ((𝜑𝜓) → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
18 addsdilem4.8 . . . . . 6 (𝜓𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
1918adantl 481 . . . . 5 ((𝜑𝜓) → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
2015, 17, 19rspcdva 3623 . . . 4 ((𝜑𝜓) → (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
2110, 20oveq12d 7449 . . 3 ((𝜑𝜓) → ((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) = (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
22 oveq1 7438 . . . . 5 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑧𝑂)))
23 oveq1 7438 . . . . . 6 (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑧𝑂) = (𝑋 ·s 𝑧𝑂))
242, 23oveq12d 7449 . . . . 5 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)))
2522, 24eqeq12d 2753 . . . 4 (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) ↔ (𝑋 ·s (𝐵 +s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂))))
2611oveq2d 7447 . . . . 5 (𝑧𝑂 = 𝑍 → (𝑋 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑍)))
27 oveq2 7439 . . . . . 6 (𝑧𝑂 = 𝑍 → (𝑋 ·s 𝑧𝑂) = (𝑋 ·s 𝑍))
2827oveq2d 7447 . . . . 5 (𝑧𝑂 = 𝑍 → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
2926, 28eqeq12d 2753 . . . 4 (𝑧𝑂 = 𝑍 → ((𝑋 ·s (𝐵 +s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) ↔ (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
30 addsdilem4.6 . . . . 5 (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
3130adantr 480 . . . 4 ((𝜑𝜓) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
3225, 29, 31, 9, 19rspc2dv 3637 . . 3 ((𝜑𝜓) → (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
3321, 32oveq12d 7449 . 2 ((𝜑𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
34 leftssno 27919 . . . . . . . . 9 ( L ‘𝐴) ⊆ No
35 rightssno 27920 . . . . . . . . 9 ( R ‘𝐴) ⊆ No
3634, 35unssi 4191 . . . . . . . 8 (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
3736, 8sselid 3981 . . . . . . 7 (𝜓𝑋 No )
3837adantl 481 . . . . . 6 ((𝜑𝜓) → 𝑋 No )
39 addsdilem4.2 . . . . . . 7 (𝜑𝐵 No )
4039adantr 480 . . . . . 6 ((𝜑𝜓) → 𝐵 No )
4138, 40mulscld 28161 . . . . 5 ((𝜑𝜓) → (𝑋 ·s 𝐵) ∈ No )
42 addsdilem4.3 . . . . . . 7 (𝜑𝐶 No )
4342adantr 480 . . . . . 6 ((𝜑𝜓) → 𝐶 No )
4438, 43mulscld 28161 . . . . 5 ((𝜑𝜓) → (𝑋 ·s 𝐶) ∈ No )
4541, 44addscld 28013 . . . 4 ((𝜑𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) ∈ No )
46 addsdilem4.1 . . . . . . 7 (𝜑𝐴 No )
4746, 39mulscld 28161 . . . . . 6 (𝜑 → (𝐴 ·s 𝐵) ∈ No )
4847adantr 480 . . . . 5 ((𝜑𝜓) → (𝐴 ·s 𝐵) ∈ No )
4946adantr 480 . . . . . 6 ((𝜑𝜓) → 𝐴 No )
50 leftssno 27919 . . . . . . . . 9 ( L ‘𝐶) ⊆ No
51 rightssno 27920 . . . . . . . . 9 ( R ‘𝐶) ⊆ No
5250, 51unssi 4191 . . . . . . . 8 (( L ‘𝐶) ∪ ( R ‘𝐶)) ⊆ No
5352, 18sselid 3981 . . . . . . 7 (𝜓𝑍 No )
5453adantl 481 . . . . . 6 ((𝜑𝜓) → 𝑍 No )
5549, 54mulscld 28161 . . . . 5 ((𝜑𝜓) → (𝐴 ·s 𝑍) ∈ No )
5648, 55addscld 28013 . . . 4 ((𝜑𝜓) → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)) ∈ No )
5745, 56addscld 28013 . . 3 ((𝜑𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) ∈ No )
5838, 54mulscld 28161 . . 3 ((𝜑𝜓) → (𝑋 ·s 𝑍) ∈ No )
5957, 41, 58subsubs4d 28124 . 2 ((𝜑𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
6045, 56, 41addsubsd 28112 . . . . 5 ((𝜑𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
6141, 44addscomd 28000 . . . . . . . 8 ((𝜑𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)))
6261oveq1d 7446 . . . . . . 7 ((𝜑𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)))
63 pncans 28102 . . . . . . . 8 (((𝑋 ·s 𝐶) ∈ No ∧ (𝑋 ·s 𝐵) ∈ No ) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
6444, 41, 63syl2anc 584 . . . . . . 7 ((𝜑𝜓) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
6562, 64eqtrd 2777 . . . . . 6 ((𝜑𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
6665oveq1d 7446 . . . . 5 ((𝜑𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) = ((𝑋 ·s 𝐶) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
6744, 48, 55adds12d 28041 . . . . 5 ((𝜑𝜓) → ((𝑋 ·s 𝐶) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) = ((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))))
6860, 66, 673eqtrd 2781 . . . 4 ((𝜑𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) = ((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))))
6968oveq1d 7446 . . 3 ((𝜑𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = (((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝑍)))
7044, 55addscld 28013 . . . 4 ((𝜑𝜓) → ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) ∈ No )
7148, 70, 58addsubsassd 28111 . . 3 ((𝜑𝜓) → (((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝑍)) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
7269, 71eqtrd 2777 . 2 ((𝜑𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
7333, 59, 723eqtr2d 2783 1 ((𝜑𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  cun 3949  cfv 6561  (class class class)co 7431   No csur 27684   L cleft 27884   R cright 27885   +s cadds 27992   -s csubs 28052   ·s cmuls 28132
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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133
This theorem is referenced by:  addsdi  28181
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