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Theorem selvffval 21670
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 21666 . . 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 4615 . . . . 5 (𝑖 = 𝐼 β†’ 𝒫 𝑖 = 𝒫 𝐼)
43adantr 481 . . . 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 6892 . . . . 5 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (Baseβ€˜(𝑖 mPoly π‘Ÿ)) = (Baseβ€˜(𝐼 mPoly 𝑅)))
7 difeq1 4114 . . . . . . . 8 (𝑖 = 𝐼 β†’ (𝑖 βˆ– 𝑗) = (𝐼 βˆ– 𝑗))
87adantr 481 . . . . . . 7 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (𝑖 βˆ– 𝑗) = (𝐼 βˆ– 𝑗))
9 simpr 485 . . . . . . 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 481 . . . . . . . . . . . 12 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (𝑖 evalSub 𝑑) = (𝐼 evalSub 𝑑))
1312fveq1d 6890 . . . . . . . . . . 11 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ ((𝑖 evalSub 𝑑)β€˜ran 𝑑) = ((𝐼 evalSub 𝑑)β€˜ran 𝑑))
1413fveq1d 6890 . . . . . . . . . 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 483 . . . . . . . . . . 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 6890 . . . . . . . . . . . . 13 ((𝑖 = 𝐼 ∧ π‘Ÿ = 𝑅) β†’ (((𝑖 βˆ– 𝑗) mVar π‘Ÿ)β€˜π‘₯) = (((𝐼 βˆ– 𝑗) mVar 𝑅)β€˜π‘₯))
1817fveq2d 6892 . . . . . . . . . . . 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 4547 . . . . . . . . . . 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 5238 . . . . . . . . . 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 6895 . . . . . . . . 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 3899 . . . . . . . 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 3899 . . . . . . 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 3899 . . . . . 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 3901 . . . . 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 5238 . . . 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 5238 . . 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 482 . 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 3494 . 2 (πœ‘ β†’ 𝐼 ∈ V)
31 selvffval.r . . 3 (πœ‘ β†’ 𝑅 ∈ π‘Š)
3231elexd 3494 . 2 (πœ‘ β†’ 𝑅 ∈ V)
3329pwexd 5376 . . 3 (πœ‘ β†’ 𝒫 𝐼 ∈ V)
3433mptexd 7222 . 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  β¦‹csb 3892   βˆ– cdif 3944  ifcif 4527  π’« cpw 4601   ↦ cmpt 5230  ran crn 5676   ∘ ccom 5679  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17140  algSccascl 21398   mVar cmvr 21449   mPoly cmpl 21450   evalSub ces 21624   selectVars cslv 21662
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-selv 21666
This theorem is referenced by:  selvfval  21671
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