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Theorem selvffval 22018
Description: Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
Hypotheses
Ref Expression
selvffval.i (πœ‘ β†’ 𝐼 ∈ 𝑉)
selvffval.r (πœ‘ β†’ 𝑅 ∈ π‘Š)
Assertion
Ref Expression
selvffval (πœ‘ β†’ (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))))
Distinct variable groups:   𝑗,𝐼,𝑓,𝑒,𝑑,𝑐,𝑑,π‘₯   𝑅,𝑗,𝑓,𝑒,𝑑,𝑐,𝑑,π‘₯
Allowed substitution hints:   πœ‘(π‘₯,𝑒,𝑑,𝑓,𝑗,𝑐,𝑑)   𝑉(π‘₯,𝑒,𝑑,𝑓,𝑗,𝑐,𝑑)   π‘Š(π‘₯,𝑒,𝑑,𝑓,𝑗,𝑐,𝑑)
Proof of Theorem selvffval
Dummy variables 𝑖 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-selv 22017 . . 3 selectVars = (𝑖 ∈ V, π‘Ÿ ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (Baseβ€˜(𝑖 mPoly π‘Ÿ)) ↦ ⦋((𝑖 βˆ– 𝑗) mPoly π‘Ÿ) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))))))
21a1i 11 . 2 (πœ‘ β†’ selectVars = (𝑖 ∈ V, π‘Ÿ ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (Baseβ€˜(𝑖 mPoly π‘Ÿ)) ↦ ⦋((𝑖 βˆ– 𝑗) mPoly π‘Ÿ) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯)))))))))
3 pweq 4611 . . . . 5 (𝑖 = 𝐼 β†’ 𝒫 𝑖 = 𝒫 𝐼)
43adantr 480 . . . 4 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ 𝒫 𝑖 = 𝒫 𝐼)
5 oveq12 7414 . . . . . 6 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (𝑖 mPoly π‘Ÿ) = (𝐼 mPoly 𝑅))
65fveq2d 6889 . . . . 5 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(𝑖 mPoly π‘Ÿ)) = (Baseβ€˜(𝐼 mPoly 𝑅)))
7 difeq1 4110 . . . . . . . 8 (𝑖 = 𝐼 β†’ (𝑖 βˆ– 𝑗) = (𝐼 βˆ– 𝑗))
87adantr 480 . . . . . . 7 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (𝑖 βˆ– 𝑗) = (𝐼 βˆ– 𝑗))
9 simpr 484 . . . . . . 7 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ π‘Ÿ = 𝑅)
108, 9oveq12d 7423 . . . . . 6 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ((𝑖 βˆ– 𝑗) mPoly π‘Ÿ) = ((𝐼 βˆ– 𝑗) mPoly 𝑅))
11 oveq1 7412 . . . . . . . . . . . . 13 (𝑖 = 𝐼 β†’ (𝑖 evalSub 𝑑) = (𝐼 evalSub 𝑑))
1211adantr 480 . . . . . . . . . . . 12 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (𝑖 evalSub 𝑑) = (𝐼 evalSub 𝑑))
1312fveq1d 6887 . . . . . . . . . . 11 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ((𝑖 evalSub 𝑑)β€˜ran 𝑑) = ((𝐼 evalSub 𝑑)β€˜ran 𝑑))
1413fveq1d 6887 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓)) = (((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓)))
15 simpl 482 . . . . . . . . . . 11 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ 𝑖 = 𝐼)
168, 9oveq12d 7423 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ((𝑖 βˆ– 𝑗) mVar π‘Ÿ) = ((𝐼 βˆ– 𝑗) mVar 𝑅))
1716fveq1d 6887 . . . . . . . . . . . . 13 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯) = (((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))
1817fveq2d 6889 . . . . . . . . . . . 12 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯)) = (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯)))
1918ifeq2d 4543 . . . . . . . . . . 11 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))) = if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))
2015, 19mpteq12dv 5232 . . . . . . . . . 10 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯)))) = (π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯)))))
2114, 20fveq12d 6892 . . . . . . . . 9 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))) = ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))
2221csbeq2dv 3895 . . . . . . . 8 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ⦋(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))) = ⦋(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))
2322csbeq2dv 3895 . . . . . . 7 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ⦋(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))) = ⦋(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))
2423csbeq2dv 3895 . . . . . 6 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ⦋(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))) = ⦋(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))
2510, 24csbeq12dv 3897 . . . . 5 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ⦋((𝑖 βˆ– 𝑗) mPoly π‘Ÿ) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))) = ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))
266, 25mpteq12dv 5232 . . . 4 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (𝑓 ∈ (Baseβ€˜(𝑖 mPoly π‘Ÿ)) ↦ ⦋((𝑖 βˆ– 𝑗) mPoly π‘Ÿ) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯)))))) = (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯)))))))
274, 26mpteq12dv 5232 . . 3 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (Baseβ€˜(𝑖 mPoly π‘Ÿ)) ↦ ⦋((𝑖 βˆ– 𝑗) mPoly π‘Ÿ) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))))) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))))
2827adantl 481 . 2 ((πœ‘ ∧ (𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅)) β†’ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (Baseβ€˜(𝑖 mPoly π‘Ÿ)) ↦ ⦋((𝑖 βˆ– 𝑗) mPoly π‘Ÿ) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝑖 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝑖 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯))))))) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))))
29 selvffval.i . . 3 (πœ‘ β†’ 𝐼 ∈ 𝑉)
3029elexd 3489 . 2 (πœ‘ β†’ 𝐼 ∈ V)
31 selvffval.r . . 3 (πœ‘ β†’ 𝑅 ∈ π‘Š)
3231elexd 3489 . 2 (πœ‘ β†’ 𝑅 ∈ V)
3329pwexd 5370 . . 3 (πœ‘ β†’ 𝒫 𝐼 ∈ V)
3433mptexd 7221 . 2 (πœ‘ β†’ (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))) ∈ V)
352, 28, 30, 32, 34ovmpod 7556 1 (πœ‘ β†’ (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Baseβ€˜(𝐼 mPoly 𝑅)) ↦ ⦋((𝐼 βˆ– 𝑗) mPoly 𝑅) / π‘’β¦Œβ¦‹(𝑗 mPoly 𝑒) / π‘‘β¦Œβ¦‹(algScβ€˜π‘‘) / π‘β¦Œβ¦‹(𝑐 ∘ (algScβ€˜π‘’)) / π‘‘β¦Œ((((𝐼 evalSub 𝑑)β€˜ran 𝑑)β€˜(𝑑 ∘ 𝑓))β€˜(π‘₯ ∈ 𝐼 ↦ if(π‘₯ ∈ 𝑗, ((𝑗 mVar 𝑒)β€˜π‘₯), (π‘β€˜(((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  β¦‹csb 3888   βˆ– cdif 3940  ifcif 4523  π’« cpw 4597   ↦ cmpt 5224  ran crn 5670   ∘ ccom 5673  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17153  algSccascl 21747   mVar cmvr 21799   mPoly cmpl 21800   evalSub ces 21975   selectVars cslv 22013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-selv 22017
This theorem is referenced by:  selvfval  22019
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